eISSN: 2577-8242 FMRIJ

Fluid Mechanics Research International Journal
Review Article
Volume 1 Issue 3 - 2017

Simulation of the Propagation of Tsunamis in Coastal Regions by a Two-Dimensional Non-Hydrostatic Shallow Water Solver
Department of Civil Engineering, University of Palermo, Italy
Received: November 14, 2017 | Published: November 27, 2017
*Corresponding author: Costanza Arico, Department of Civil Engineering, Environmental, Aerospace, Materials, University of Palermo, Avenue of Sciences, 90128, Palermo, Italy, Email:
Citation: Arico C (2017) Simulation of the Propagation of Tsunamis in Coastal Regions by a Two-Dimensional Non-Hydrostatic Shallow Water Solver. Fluid Mech Res Int 1(3): 00011. DOI: 10.15406/fmrij.2017.01.00011

Abstract

Due to the enormous damages and losses of human lives in the inundated regions, the simulation of the propagation of tsunamis in coastal areas has received an increasing interest of the researchers. We present a 2D depth-integrated, non-hydrostatic shallow waters solver to simulate the propagation of tsunamis, solitary waves and surges in coastal regions. We write the governing continuity and momentum equations in conservative form and discretize the domain with unstructured triangular Generalized Delaunay meshes. We apply a fractional-time-step procedure, where two problems (steps) are consecutively solved. In the first and in the second step, we hypothesize a hydrostatic and a non-hydrostatic distribution of the pressure, respectively. Several literature models, which solve the same set of equations, are based on a fractional-time-step procedure. In the hydrostatic step of these literature solvers, the flow field does not satisfy the depth-integrated continuity equation, since the momentum equations are solved independently of the continuity equation. In both steps of the proposed numerical scheme, the computed flow field satisfies the depth-integrated continuity equation discretized in each computational cell. This is obtained by solving together the governing equations, according to the different numerical procedures adopted in the steps of the algorithm. Wet/dry problems are implicitly embedded in the proposed numerical solver. We present several model applications where analytical or measured reference solutions are available. The computational effort required by the proposed procedure is investigated.

Keywords: Shallow waters; Non-hydrostatic pressure; Unstructured mesh; Wetting/drying; Tsunami propagation; Long waves; Voronoi cells; Runge-Kutta method; Galerkin scheme; Manning equation; Dirichlet condition; OpenFOAM

Abbrevations

BTMs: Boussinesq-Type Models; NLSWEs: Nonlinear Shallow Water Equations; FV: Finite Volumes; WD: Wetting/Drying; PDEs: Partial Differential Equations; DC: Diffusive Correction; MAST: Marching in Space and Time; CC: Convective Correction; GD: Generalized Delaunay; ODEs: Ordinary Differential Equations; CP: Convective Prediction

Introduction

During the last years, great effort has been addressed by several authors to simulate the propagation of solitary waves/tsunamis, tides or surges, due to the tremendous damages and losses of human lives in the inundated rural and residential areas. Tsunamis are sea waves usually generated by undersea landslides and earthquakes. They can be regarded as long/solitary waves with small amplitude and long wavelength, travelling with high speed over long distances. Approaching the coast, their amplitude increases, becoming potentially destructive. The propagation of tsunami in coastal regions can be studied by investigating the shoaling and breaking of solitary waves over inclined bottoms [1]. Chanson [2] asserted that the front of tsunamis over dry plains becomes a shock wave, and presented a similarity between the propagation of the tsunamis over dry coastal areas and the classical dam-break problems. Generally, 3D simulations of the processes described as above, e.g., by RANS models [3], or Smoothed Particle Hydrodynamics methods [4], or Volume of Fluids methods [5], require very high computational costs. For these reasons, a significant amount of literature based on depth-integrated equations has been published for simulations of long waves/tsunamis propagation. Generally, the two modeling approaches proposed in literature are the Boussinesq-Type Models (BTMs) and the Non-Hydrostatic Nonlinear Shallow Water Equations (NLSWEs) models.

The classical formulation of the BTMs [6] involves weak nonlinearity and dispersion. High-order BTMs have been proposed to overcome these difficulties, but suffer from complex numerical discretization, which leads to high computational efforts [7,8]. Generally, the BTMs also suffer from the inclusion, in the governing equations, of extra terms, based on empirical considerations, introduced for the simulation of wave breaking with the associated energy dissipation, and wetting/drying transition [8].

Hereafter, we call "hydrostatic NLSWEs" and "non-hydrostatic NLSWEs" (or "dynamic NLSWEs") respectively the NLSWEs with hydrostatic and non-hydrostatic (or dynamic) pressure distribution. Due to simplicity and accuracy, the hydrostatic NLSWEs, are largely applied for simulating wave processes in rivers, lakes, estuaries. The hydrostatic NLSWEs lack frequency dispersion and are not able in simulating some dispersive features of long waves/tsunamis propagation [8-13] as well as free-surface undulations at front/tail of bores and shock waves of tsunamis and dam-breaks [14,15]. Essentially two aspects allow reproducing the dispersive effects in the NLSWEs, 1) the decomposition of the pressure into its hydrostatic and dynamic components and 2) the inclusion of the vertical momentum equation in the governing equations [16]. Two fractional-step methodologies are usually applied to perform the decomposition of the pressure, the pressure projection and the pressure correction method, respectively [17].

One of the most demanding topics of the NLSWEs models is the simulation of the wave breaking [18]. As widely recognized, an accurate simulation of discontinuous flows (e.g., bores or jumps), is obtained if the numerical model solves the weak form of the hydrostatic NLSWEs written in conservative form [19-21]. Numerous shock-capturing procedures, in the framework of the Finite Volumes (FV), Finite Elements (FE) and Finite Differences (FD) schemes, have been proposed for the solution of the hydrostatic NLSWEs [22,23]. Only a few non-hydrostatic models are equipped with this shock-capturing feature [18,24,25].

In the NLSWEs applications involving fast transitions between wet and dry surfaces, e.g., coastal flooding and run-up of tsunamis, an accurate modeling of wetting/drying (WD) processes plays an important role. Several WD numerical techniques have been proposed [26-28], but generally these are affected by several drawbacks, e.g., the mass imbalance and the high computational costs due to additional relationships to include in the numerical procedure at the transition wet/dry surface [27].

In the present paper, we propose a non-hydrostatic NLSWEs solver to simulate the propagation of long waves, tsunamis and surges processes. We write the governing equations in conservative form and the total pressure is decomposed into its hydrostatic and dynamic components. A hydrostatic and a non-hydrostatic (or dynamic) problem (or step), consecutively solved, are embedded in a general fractional-time-step procedure. The domain is discretized with unstructured triangular Generalized Delaunay meshes [4]. Generally, the triangular meshes are suitable to fit irregular natural boundaries and arbitrary geometries. The hydrostatic step, where the dynamic pressure terms are neglected, is solved by a prediction-correction procedure, recently proposed in [30-33]. Unlike studies in [30-33], where a cell-centered formulation is applied, in the present solver we adopt a node-centered formulation.

We organize the paper as follows. We present the governing equations and the general formulation of the numerical solver in sections 2 and 3, respectively. The hydrostatic and dynamic steps are disentangled in sections 4 and 5, respectively, and the boundary conditions, in section 6. The properties of the proposed model (e.g. the proof of the local and global mass balance, the WD properties, the study of the phase speed, ... ) are outlined in section 7. Finally, in section 8, we present several model applications where reference analytical solutions or measured data are available. The proposed test cases highlight the skill of the model in simulating challenging features of the propagation of long, solitary waves and tsunami, as the wave breaking, run-up, and wetting/drying problems. In the same section we also investigate the computational burden.

The Governing Equations

We derive the governing equations from the continuity and Reynolds equations of an incompressible fluid (water),

u=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqGHhi s0cqGHflY1caWH1bGaeyypa0JaaGimaaaa@3D12@                                                                                 (1),

u t +( uu )+ 1 ρ p+gzυ 2 u=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyRaaCyDaaGcbaqcLbsacqGHciITcaWG0baaaiab gUcaRiabgEGirlabgwSixNqbaoaabmaakeaajugibiaahwhacaWH1b aakiaawIcacaGLPaaajugibiabgUcaRKqbaoaalaaakeaajugibiac aImIXaaakeaajugibiadaIdHbpGCaaGaey4bIeTaamiCaiabgUcaRi aahEgacqGHflY1cqGHhis0caWG6bGaeyOeI0IaeqyXduNaey4bIeDc fa4aaWbaaSqabKqaGeaajugWaiaaikdaaaqcLbsacaWH1bGaeyypa0 JaaGimaaaa@612E@                          (2),

where t is the time, u is the velocity vector (u, v, and w are its components along x, y and z directions, respectively), g is the gravitational acceleration vector, g=  ( 0 0g ) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWHNbGaeyypa0JaaeiiaOWdamaabmaabaqcLbsapeGaaGim aiaabccacaaIWaGaam4zaaGcpaGaayjkaiaawMcaamaaCaaaleqaje aibaqcLbmapeGaamivaaaaaaa@4159@ , with g the norm of the gravitational acceleration (notation (*)T indicates the transposed of vector (*)), ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHbpGCaaa@38B5@  is the density of the water, p is the pressure and υ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHfpqDaaa@38BC@ is the kinematic viscosity coefficient.

We assume a physical domain bounded along the vertical directions by the free surface H(x, y, t) and the bed surface (or bottom) zb(x, y, t), and h is the water depth (h = H - zb) (Figure 1). We operate under the hypothesis of fixed bed condition, i.e., z b / t =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSGbaO qaaKqzGeGaeyOaIyRaamOEaKqbaoaaBaaajqwaa+FaaKqzadGaamOy aaWcbeaaaOqaaKqzGeGaeyOaIyRaamiDaaaacqGH9aqpcaaIWaaaaa@430B@ , which implies H/ t = h/ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqqFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSGbaO qaaKqzGeGaeyOaIyRaamisaaGcbaqcLbsacqGHciITcaWG0baaaiab g2da9KqbaoaalyaakeaajugibiabgkGi2kaadIgaaOqaaKqzGeGaey OaIyRaamiDaaaaaaa@43AB@ . The total derivatives of the bottom and free surfaces lead to the kinematic boundary conditions [34]

w b = D z b Dt = u b z b x + v b z b y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWG3b qcfa4aaSbaaKqaGeaajugWaiaadkgaaSqabaqcLbsacqGH9aqpjuaG daWcaaGcbaqcLbsacaWGebGaamOEaKqbaoaaBaaajeaibaqcLbmaca WGIbaaleqaaaGcbaqcLbsacaWGebGaamiDaaaacqGH9aqpcaWG1bqc fa4aaSbaaKqaGeaajugWaiaadkgaaSqabaqcfa4aaSaaaOqaaKqzGe GaeyOaIyRaamOEaKqbaoaaBaaajeaibaqcLbmacaWGIbaaleqaaaGc baqcLbsacqGHciITcaWG4baaaiabgUcaRiaadAhajuaGdaWgaaqcba saaKqzadGaamOyaaWcbeaajuaGdaWcaaGcbaqcLbsacqGHciITcaWG 6bqcfa4aaSbaaKqaGeaajugWaiaadkgaaSqabaaakeaajugibiabgk Gi2kaadMhaaaaaaa@6153@  and w s = DH Dt = H t + u s H x + v s H y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWG3b qcfa4aaSbaaKqaGeaajugWaiaadohaaSqabaqcLbsacqGH9aqpjuaG daWcaaGcbaqcLbsacaWGebGaamisaaGcbaqcLbsacaWGebGaamiDaa aacqGH9aqpjuaGdaWcaaGcbaqcLbsacqGHciITcaWGibaakeaajugi biabgkGi2kaadshaaaGaey4kaSIaamyDaKqbaoaaBaaajeaibaqcLb macaWGZbaaleqaaKqbaoaalaaakeaajugibiabgkGi2kaadIeaaOqa aKqzGeGaeyOaIyRaamiEaaaacqGHRaWkcaWG2bqcfa4aaSbaaKqaGe aajugWaiaadohaaSqabaqcfa4aaSaaaOqaaKqzGeGaeyOaIyRaamis aaGcbaqcLbsacqGHciITcaWG5baaaaaa@5F49@           (3),

and the sub-indices b and s mark the values at the bottom and at the free surface, respectively. The (total) pressure p is split into its hydrostatic and dynamic components [16],

p( x,t )= ρg( H( x,t )z ) hydrostatic component + q ^ ( x,t ) dynamic component       x= ( x y z ) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGWb qcfa4aaeWaaOqaaKqzGeGaaCiEaiaacYcacaWG0baakiaawIcacaGL Paaajugibiabg2da9Kqbaoaayaaakeaajugibiabeg8aYjaadEgaju aGdaqadaGcbaqcLbsacaWGibqcfa4aaeWaaOqaaKqzGeGaaCiEaiaa cYcacaWG0baakiaawIcacaGLPaaajugibiabgkHiTiaadQhaaOGaay jkaiaawMcaaaqcbasaaKqzadGaaeiAaiaabMhacaqGKbGaaeOCaiaa b+gacaqGZbGaaeiDaiaabggacaqG0bGaaeyAaiaabogacaqGGaGaae 4yaiaab+gacaqGTbGaaeiCaiaab+gacaqGUbGaaeyzaiaab6gacaqG 0baakiaawIJ=aKqzGeGaey4kaSscfa4aaGbaaOqaaKqzGeGabmyCay aajaqcfa4aaeWaaOqaaKqzGeGaaCiEaiaacYcacaWG0baakiaawIca caGLPaaaaKqaGeaajugWaiaabsgacaqG5bGaaeOBaiaabggacaqGTb GaaeyAaiaabogacaqGGaGaae4yaiaab+gacaqGTbGaaeiCaiaab+ga caqGUbGaaeyzaiaab6gacaqG0baakiaawIJ=aKqzGeGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaCiEaiabg2da9Kqbaoaabmaa keaajugibuaabeqabmaaaOqaaKqzGeGaamiEaaGcbaqcLbsacaWG5b aakeaajugibiaadQhaaaaakiaawIcacaGLPaaajuaGdaahaaWcbeqc KfaG=haajugOaiaadsfaaaaaaa@92B6@                                         (4).

Figure 1: Definition sketch.

At z = H we set p= q ^ = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGWbGaeyypa0JabmyCayaajaGaeyypa0Jaaeiiaiaaicda aaa@3C59@ , i.e., zero value of the atmospheric pressure.

If we integrate Eqs. (1)-(2) over the water depth h, from zb to H, we derive the depth-integrated NLSWEs [35]. If we merge Eqn (3) in the continuity equation, integrated along the water depth, it results

h t + ( Uh ) x + ( Vh ) y =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaK aaGfaajugibiabgkGi2kaadIgaaKaaGfaajugibiabgkGi2kaadsha aaGaey4kaSscfa4aaSaaaKaaGfaajugibiabgkGi2Mqbaoaabmaaja aybaqcLbsacaWGvbGaamiAaaqcaaMaayjkaiaawMcaaaqaaKqzGeGa eyOaIyRaamiEaaaacqGHRaWkjuaGdaWcaaqcaawaaKqzGeGaeyOaIy Bcfa4aaeWaaKaaGfaajugibiaadAfacaWGObaajaaycaGLOaGaayzk aaaabaqcLbsacqGHciITcaWG5baaaiabg2da9iaaicdaaaa@5715@                                                  (5).

If we hypothesize that q ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqqFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGXb GbaKaaaaa@374A@  changes linearly along z, from value 0 at z = H to value q ^ b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGXb GbaKaajuaGdaWgaaqcbasaaKqzadGaamOyaaWcbeaaaaa@3A83@  at z = zb, dividing Eq. (2) by ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHbpGCaaa@38B5@ , the depth-integrated momentum equations are

( Uh ) t local derivative term + ( U 2 h ) x + ( UVh ) y convective terms + gh h x x-component  of depth-integrated hydrostatic pressure + gh z b x x-component  of hydrostatic pressure at bottm + 1 2 ( q b h ) x x-component  of depth-integrated dynamic pressure + q b z b x x-component  of dynamic pressure at bottm + g n 2 ( Uh ) ( Uh ) 2 + ( Vh ) 2 h 7/3 x-component  of bottom stress =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGceaGabeaajuaGda agaaGcbaqcfa4aaSaaaOqaaKqzGeGaeyOaIyRaaGjcVNqbaoaabmaa keaajugibiaadwfacaWGObaakiaawIcacaGLPaaaaeaajugibiabgk Gi2kaadshaaaaaeaGabSqaaKqzadGaaeiBaiaab+gacaqGJbGaaeyy aiaabYgacaqGGaGaaeizaiaabwgacaqGYbGaaeyAaiaabAhacaqGHb GaaeiDaiaabMgacaqG2bGaaeyzaaWcbaqcLbmacaqG0bGaaeyzaiaa bkhacaqGTbaaaOGaayjo+dqcLbsacaaMc8Uaey4kaSIaaGPaVNqbao aayaaakeaajuaGdaWcaaGcbaqcLbsacqGHciITcaaMi8Ecfa4aaeWa aOqaaKqzGeGaamyvaKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaK qzGeGaamiAaaGccaGLOaGaayzkaaaabaqcLbsacqGHciITcaWG4baa aiaayIW7caaMc8UaaCjaVlabgUcaRiaaykW7juaGdaWcaaGcbaqcLb sacqGHciITcaaMi8Ecfa4aaeWaaOqaaKqzGeGaamyvaiaadAfacaWG ObaakiaawIcacaGLPaaaaeaajugibiabgkGi2kaadMhaaaaajeaiba qcLbmacaqGJbGaae4Baiaab6gacaqG2bGaaeyzaiaabogacaqG0bGa aeyAaiaabAhacaqGLbGaaeiiaiaabshacaqGLbGaaeOCaiaab2gaca qGZbaakiaawIJ=aKqzGeGaaGjcVlaaykW7cqGHRaWkjuaGdaagaaGc baqcLbsacaWGNbGaamiAaKqbaoaalaaakeaajugibiabgkGi2kaadI gaaOqaaKqzGeGaeyOaIyRaamiEaaaaaqaaceWcbaqcLbmacaWG4bGa aeylaiaabogacaqGVbGaaeyBaiaabchacaqGVbGaaeOBaiaabwgaca qGUbGaaeiDaiaabccaaKqaGeaajugWaiaab+gacaqGMbGaaeiiaiaa bsgacaqGLbGaaeiCaiaabshacaqGObGaaeylaiaabMgacaqGUbGaae iDaiaabwgacaqGNbGaaeOCaiaabggacaqG0bGaaeyzaiaabsgaaSqa aKqzadGaaeiAaiaabMhacaqGKbGaaeOCaiaab+gacaqGZbGaaeiDai aabggacaqG0bGaaeyAaiaabogacaqGGaGaaeiCaiaabkhacaqGLbGa ae4CaiaabohacaqG1bGaaeOCaiaabwgaaaGccaGL44pajugibiabgU caRKqbaoaayaaakeaajugibiaadEgacaWGObqcfa4aaSaaaOqaaKqz GeGaeyOaIyRaamOEaKqbaoaaBaaajeaibaqcLbmacaWGIbaaleqaaa GcbaqcLbsacqGHciITcaWG4baaaaabaiqaleaajugWaiaadIhacaqG TaGaae4yaiaab+gacaqGTbGaaeiCaiaab+gacaqGUbGaaeyzaiaab6 gacaqG0bGaaeiiaaqcbasaaKqzadGaae4BaiaabAgacaqGGaGaaeiA aiaabMhacaqGKbGaaeOCaiaab+gacaqGZbGaaeiDaiaabggacaqG0b GaaeyAaiaabogaaSqaaKqzadGaaeiCaiaabkhacaqGLbGaae4Caiaa bohacaqG1bGaaeOCaiaabwgacaqGGaGaaeyyaiaabshacaqGGaGaae Oyaiaab+gacaqG0bGaaeiDaiaab2gaaaGccaGL44pajugibiaaykW7 cqGHRaWkjuaGdaagaaGcbaqcfa4aaSaaaOqaaKqzGeGaaGymaaGcba qcLbsacaaIYaaaaiaaykW7juaGdaWcaaGcbaqcLbsacqGHciITcaaM b8Ecfa4aaeWaaOqaaKqzGeGaamyCaKqbaoaaBaaajeaibaqcLbmaca WGIbaaleqaaKqzGeGaamiAaaGccaGLOaGaayzkaaaabaqcLbsacqGH ciITcaWG4baaaaabaiqaleaajugWaiaadIhacaqGTaGaae4yaiaab+ gacaqGTbGaaeiCaiaab+gacaqGUbGaaeyzaiaab6gacaqG0bGaaeii aaqcbasaaKqzadGaae4BaiaabAgacaqGGaGaaeizaiaabwgacaqGWb GaaeiDaiaabIgacaqGTaGaaeyAaiaab6gacaqG0bGaaeyzaiaabEga caqGYbGaaeyyaiaabshacaqGLbGaaeizaaWcbaqcLbmacaqGKbGaae yEaiaab6gacaqGHbGaaeyBaiaabMgacaqGJbGaaeiiaiaabchacaqG YbGaaeyzaiaabohacaqGZbGaaeyDaiaabkhacaqGLbaaaOGaayjo+d qcLbsacaaMc8Uaey4kaSscfa4aaGbaaOqaaKqzGeGaamyCaKqbaoaa BaaajeaibaqcLbmacaWGIbaaleqaaKqzGeGaaGjcVNqbaoaalaaake aajugibiabgkGi2kaadQhajuaGdaWgaaqcbasaaKqzadGaamOyaaWc beaaaOqaaKqzGeGaeyOaIyRaamiEaaaaaqaaceWcbaqcLbmacaWG4b GaaeylaiaabogacaqGVbGaaeyBaiaabchacaqGVbGaaeOBaiaabwga caqGUbGaaeiDaiaabccaaKqaGeaajugWaiaab+gacaqGMbGaaeiiai aabsgacaqG5bGaaeOBaiaabggacaqGTbGaaeyAaiaabogaaSqaaKqz adGaaeiCaiaabkhacaqGLbGaae4CaiaabohacaqG1bGaaeOCaiaabw gacaqGGaGaaeyyaiaabshacaqGGaGaaeOyaiaab+gacaqG0bGaaeiD aiaab2gaaaGccaGL44pajugibiaaykW7cqGHRaWkaOqaaKqzGeGaaG PaVNqbaoaayaaakeaajuaGdaWcaaGcbaqcLbsacaWGNbGaamOBaKqb aoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaaGzaVNqbaoaabm aakeaajugibiaadwfacaWGObaakiaawIcacaGLPaaajugibiaayIW7 juaGdaGcaaGcbaqcfa4aaeWaaOqaaKqzGeGaamyvaiaadIgaaOGaay jkaiaawMcaaKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGa aGjcVlabgUcaRKqbaoaabmaakeaajugibiaadAfacaWGObaakiaawI cacaGLPaaajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaSqabaaa keaajugibiaadIgajuaGdaahaaWcbeqcbasaaKqzadGaaG4naiaac+ cacaaIZaaaaaaaaKqzGeabaiqaleaajugWaiaadIhacaqGTaGaae4y aiaab+gacaqGTbGaaeiCaiaab+gacaqGUbGaaeyzaiaab6gacaqG0b GaaeiiaaWcbaqcLbmacaqGVbGaaeOzaiaabccacaqGIbGaae4Baiaa bshacaqG0bGaae4Baiaab2gacaqGGaGaae4CaiaabshacaqGYbGaae yzaiaabohacaqGZbaaaOGaayjo+dqcLbsacqGH9aqpcaaIWaaaaaa@E122@                          (6.a),

( Vh ) t + ( UVh ) x + ( V 2 h ) y +gh h y +gh z b y + 1 2 ( q b h ) y + q b z b y + g n 2 ( Vh ) ( Uh ) 2 + ( Vh ) 2 h 7/3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyRaaGzaVNqbaoaabmaakeaajugibiaadAfacaWG ObaakiaawIcacaGLPaaaaeaajugibiabgkGi2kaadshaaaGaaGPaVl abgUcaRiaaykW7juaGdaWcaaGcbaqcLbsacqGHciITcaaMb8Ecfa4a aeWaaOqaaKqzGeGaamyvaiaadAfacaWGObaakiaawIcacaGLPaaaae aajugibiabgkGi2kaadIhaaaGaaGPaVlabgUcaRKqbaoaalaaakeaa jugibiabgkGi2kaaygW7juaGdaqadaGcbaqcLbsacaWGwbqcfa4aaW baaSqabKqaGeaajugWaiaaikdaaaqcLbsacaWGObaakiaawIcacaGL PaaaaeaajugibiabgkGi2kaadMhaaaGaaGPaVlabgUcaRiaadEgaca WGObqcfa4aaSaaaOqaaKqzGeGaeyOaIyRaamiAaaGcbaqcLbsacqGH ciITcaWG5baaaiaaykW7cqGHRaWkcaWGNbGaamiAaKqbaoaalaaake aajugibiabgkGi2kaadQhajuaGdaWgaaqcbasaaKqzadGaamOyaaWc beaaaOqaaKqzGeGaeyOaIyRaamyEaaaacaaMc8Uaey4kaSIaaGPaVN qbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaaGOmaaaajuaGdaWc aaGcbaqcLbsacqGHciITcaaMb8Ecfa4aaeWaaOqaaKqzGeGaamyCaK qbaoaaBaaaleaajugibiaadkgaaSqabaqcLbsacaWGObaakiaawIca caGLPaaaaeaajugibiabgkGi2kaadMhaaaGaaGPaVlabgUcaRiaadg hajuaGdaWgaaWcbaqcLbsacaWGIbaaleqaaKqzGeGaaGPaVNqbaoaa laaakeaajugibiabgkGi2kaadQhajuaGdaWgaaqcbasaaKqzadGaam OyaaWcbeaaaOqaaKqzGeGaeyOaIyRaamyEaaaacqGHRaWkcaaMc8Ec fa4aaSaaaOqaaKqzGeGaam4zaiaad6gajuaGdaahaaWcbeqcbasaaK qzadGaaGOmaaaajugibiaayIW7juaGdaqadaGcbaqcLbsacaWGwbGa amiAaaGccaGLOaGaayzkaaqcLbsacaaMb8Ecfa4aaOaaaOqaaKqbao aabmaakeaajugibiaadwfacaWGObaakiaawIcacaGLPaaajuaGdaah aaWcbeqcbasaaKqzadGaaGOmaaaajugibiabgUcaRKqbaoaabmaake aajugibiaadAfacaWGObaakiaawIcacaGLPaaajuaGdaahaaWcbeqc basaaKqzadGaaGOmaaaaaSqabaaakeaajugibiaadIgajuaGdaahaa WcbeqcbasaaKqzadGaaG4naiaac+cacaaIZaaaaaaajugibiabg2da 9iaaicdaaaa@CCF9@               (6.b),

( Wh ) t = q b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyBcfa4aaeWaaOqaaKqzGeGaam4vaiaadIgaaOGa ayjkaiaawMcaaaqaaKqzGeGaeyOaIyRaamiDaaaacqGH9aqpcaWGXb qcfa4aaSbaaKqaGeaajugWaiaadkgaaSqabaaaaa@44F7@                         (6,c),

and the components of the depth-integrated velocity vector along x, y and z directionsare U, V and W, respectively, and the corresponding components of the specific flow rate vector (i.e. the flow rate per unitary width) are Uh, Vh and Wh, q b = q ^ b /ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGXb qcfa4aaSbaaKqaGeaajugWaiaadkgaaSqabaqcLbsacqGH9aqpjuaG daWcgaGcbaqcLbsaceWGXbGbaKaajuaGdaWgaaqcbasaaKqzadGaam OyaaWcbeaaaOqaaKqzGeGaeqyWdihaaaaa@439D@  and n is the Manning bed roughness coefficient. The Leibniz rule is applied to perform the integration along the depth of the terms proportional to qb [11,18]. We refer the reader to [36] for further details of the depth-integration of the governing equations. For brevity, we only explicit the terms in the x-momentum equation (6.a), and the terms in the other two momentum equations have similar meanings along the y and z directions, respectively. In x and y momentum equations (Eqs. (6.a)-(6.b)), we neglect the stress at the free surface (e.g., due to the action of the wind), and the depth-averaged turbulent/viscous stresses. Similarly to other studies [8,11,13,17,37,38], in Eq. (6.c), we neglect the vertical advection and dissipation terms compared with the dynamic pressure term, that is, we assume that the dynamic pressure is essentially due to the local time derivative of the vertical velocity of the water column.

The linear vertical variation of qb has been adopted by other authors [8,11,13,17,18,37,38] and it is consistent with the hypothesis of zero value of the atmospheric pressure. Following other authors, 8,11,13,17,18,37,38] we also hypothesize that the vertical component of the velocity changes linearly, from wb to ws, and its depth-integrated W value is

W= ( w s + w b )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGxb Gaeyypa0tcfa4aaSGbaOqaaKqbaoaabmaakeaajugibiaadEhajuaG daWgaaqcbasaaKqzadGaam4CaaWcbeaajugibiabgUcaRiaadEhaju aGdaWgaaqcbasaaKqzadGaamOyaaWcbeaaaOGaayjkaiaawMcaaaqa aKqzGeGaaGOmaaaaaaa@4685@                               (7),

Both assumptions of the vertical distribution of qb and w require further investigation, but beyond the purpose of this paper. The unknown variables of the governing Partial Differential Equations (PDEs) (5)-(6) are h, Uh, Vh, Wh and qb.

General Features of the Numerical Algorithm

The fractional-time-step procedure

A fractional-time-step scheme, in the general framework of the pressure projection methods [17], is applied to solve the governing equations (5)-(6). The hydrostatic problem, in its turn, is solved applying a predictor-corrector scheme. If we neglect in Eqn. (6.a)-(6.b) the terms with the dynamic pressure, the hydrostatic NLSWEs become:

h t + ( Uh ) x + ( Vh ) y =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyRaamiAaaGcbaqcLbsacqGHciITcaWG0baaaiab gUcaRKqbaoaalaaakeaajugibiabgkGi2Mqbaoaabmaakeaajugibi aadwfacaWGObaakiaawIcacaGLPaaaaeaajugibiabgkGi2kaadIha aaGaey4kaSscfa4aaSaaaOqaaKqzGeGaeyOaIyBcfa4aaeWaaOqaaK qzGeGaamOvaiaadIgaaOGaayjkaiaawMcaaaqaaKqzGeGaeyOaIyRa amyEaaaacqGH9aqpcaaIWaaaaa@541D@                                                                                   (8),

( Uh ) t + x ( U 2 h )+ y ( UVh )+gh H x +g n 2 ( Uh ) ( Uh ) 2 + ( Vh ) 2 h 7/3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyBcfa4aaeWaaOqaaKqzGeGaamyvaiaadIgaaOGa ayjkaiaawMcaaaqaaKqzGeGaeyOaIyRaamiDaaaacqGHRaWkjuaGda WcaaGcbaqcLbsacqGHciITaOqaaKqzGeGaeyOaIyRaamiEaaaajuaG daqadaGcbaqcLbsacaWGvbqcfa4aaWbaaSqabKqaGeaajugWaiaaik daaaqcLbsacaWGObaakiaawIcacaGLPaaajugibiabgUcaRKqbaoaa laaakeaajugibiabgkGi2cGcbaqcLbsacqGHciITcaWG5baaaKqbao aabmaakeaajugibiaadwfacaWGwbGaamiAaaGccaGLOaGaayzkaaqc LbsacqGHRaWkcaWGNbGaamiAaKqbaoaalaaakeaajugibiabgkGi2k aadIeaaOqaaKqzGeGaeyOaIyRaamiEaaaacqGHRaWkcaWGNbqcfa4a aSaaaOqaaKqzGeGaamOBaKqbaoaaCaaaleqajeaibaqcLbmacaaIYa aaaKqbaoaabmaakeaajugibiaadwfacaWGObaakiaawIcacaGLPaaa juaGdaGcaaGcbaqcfa4aaeWaaOqaaKqzGeGaamyvaiaadIgaaOGaay jkaiaawMcaaKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGa ey4kaSscfa4aaeWaaOqaaKqzGeGaamOvaiaadIgaaOGaayjkaiaawM caaKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaaWcbeaaaOqaaKqz GeGaamiAaKqbaoaaCaaaleqajeaibaqcLbmacaaI3aGaai4laiaaio daaaaaaKqzGeGaeyypa0JaaGimaaaa@8898@  (9.a),

( Vh ) t + x ( UVh )+ y ( V 2 h )+gh H y +g n 2 ( Vh ) ( Uh ) 2 + ( Vh ) 2 h 7/3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyBcfa4aaeWaaOqaaKqzGeGaamOvaiaadIgaaOGa ayjkaiaawMcaaaqaaKqzGeGaeyOaIyRaamiDaaaacqGHRaWkjuaGda WcaaGcbaqcLbsacqGHciITaOqaaKqzGeGaeyOaIyRaamiEaaaajuaG daqadaGcbaqcLbsacaWGvbGaamOvaiaadIgaaOGaayjkaiaawMcaaK qzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGaeyOaIylakeaajugibiab gkGi2kaadMhaaaqcfa4aaeWaaOqaaKqzGeGaamOvaKqbaoaaCaaale qajeaibaqcLbmacaaIYaaaaKqzGeGaamiAaaGccaGLOaGaayzkaaqc LbsacqGHRaWkcaWGNbGaamiAaKqbaoaalaaakeaajugibiabgkGi2k aadIeaaOqaaKqzGeGaeyOaIyRaamyEaaaacqGHRaWkcaWGNbqcfa4a aSaaaOqaaKqzGeGaamOBaKqbaoaaCaaaleqajeaibaqcLbmacaaIYa aaaKqbaoaabmaakeaajugibiaadAfacaWGObaakiaawIcacaGLPaaa juaGdaGcaaGcbaqcfa4aaeWaaOqaaKqzGeGaamyvaiaadIgaaOGaay jkaiaawMcaaKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGa ey4kaSscfa4aaeWaaOqaaKqzGeGaamOvaiaadIgaaOGaayjkaiaawM caaKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaaWcbeaaaOqaaKqz GeGaamiAaKqbaoaaCaaaleqajeaibaqcLbmacaaI3aGaai4laiaaio daaaaaaKqzGeGaeyypa0JaaGimaaaa@889C@                      (9.b),

and the vertical momentum equation becomes trivial. The unknown variables of system (8)-(9) are h, Uh and Vh. In Appendix A we give more details of the predictor-corrector procedure applied for the solution of the hydrostatic problem (8)-(9). The functional characteristics of the prediction and correction steps of problem (8)-(9) are the ones of a convective and a diffusive process, respectively [31,32] (see also Appendix A). According to this, the prediction and correction systems are named "convective prediction" (CP0) and "diffusive correction" (DC) system, respectively. The original hyperbolic shallow waters problem is split into a convective plus a diffusive one.

We solve the hydrostatic CP0 problem using the A Marching in Space and Time (MAST) procedure [30,31,33], where the cells are ordered at the beginning of each time iteration, according to rules indicated as follows, and consecutively solved. Originally, MAST has been used for scalar potential flow fields (where the flow vector and the spatial gradient of the potential have opposite signs). To adopt this procedure in cases where a scalar potential of the flow field does not exist, as the present one (where the velocity vector is an unknown of the problem), a further convective correction step has been added [31,32]. The original hydrostatic CP0 step is split into a "convective prediction" (CP) and a "convective correction" (CC) step [31,32]. We use an auxiliary scalar function, the "approximated potential" for cells ordering (details in [31,32]).

In the hydrostatic DC step, a large linear system, whose dimension is equal to the number of the computational cells, is solved for the H unknowns, and the horizontal components of the flow rate are subsequently updated. In the dynamic problem of the general fractional-time-step scheme, we merge the terms of the momentum equations proportional to the non-hydrostatic pressure, as well as the boundary conditions at the free and bottom surfaces (Eqs. (3)), in the continuity equation (Eq. (1)), integrated over a computational cell. The resulting system is solved for the qb unknowns. Hence, the components of the flow rate are corrected, while the water level (depth) is not updated.

Others numerical schemes have been recently proposed for the solution of the non-hydrostatic NLSWEs, where a fractional-time-step procedure is applied [13,18,37,39]. In the hydrostatic step of these models, the flow field does not satisfy the depth-integrated continuity equation. This is why the momentum and the continuity equations are separately solved. Unlike these literature solvers, in both the hydrostatic and dynamic steps of the present scheme, the computed flow field satisfies the depth-integrated continuity equation discretized in each cell. As shown in sections 4 and 5, this is due 1) to the simultaneous solution of the continuity and momentum equations in the CP and CC steps, and 2) to the embedding of the momentum equations in the continuity equation in the hydrostatic DC and dynamic steps. As a result, the model preserves in all its steps the local and global mass balance, as shown in the following sections.

Computational mesh properties

To facilitate the reader, we briefly remind some basic concepts and nomenclatures of the unstructured triangular Generalized Delaunay (GD) meshes [29] adopted in the present study to discretize the domain. Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqqHPoWvaaa@3883@  is a 2D bounded domain and Th an unstructured triangulation of Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqqHPoWvaaa@3883@ . NTand N represent the total number of triangles and nodes (or vertices) of Th, respectively, and Te, e = 1, …, NT, is a generic triangle of Th, with area | T e |. r i ,ip MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabaqaamaabaabaaGcbaqcfa4aaqWaaO qaaKqzGeGaamivaKqbaoaaBaaajqwaa+FaaKqzadGaamyzaaWcbeaa aOGaay5bSlaawIa7aKqzGeGaaiOlaiaaykW7caaMc8UaaGPaVlaayk W7caaMc8oeaaaaaaaaa8qacaWHYbqcfa4damaaBaaajeaibaqcLbma peGaamyAaaWcpaqabaqcfa4aaSbaaKqaGeaajugWa8qacaGGSaGaam yAaiaadchaaSWdaeqaaaaa@51E4@ is the edge vector connecting nodes i and ip, oriented from i to ip. Triangle Te shares side ri,ip with Tep, c i,ip T q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiaado galmaaDaaajeaibaqcLbmacaWGPbGaaiilaiaadMgacaWGWbaajeai baqcLbmacaWGubWcdaWgaaqccasaaKqzGcGaamyCaaqccasabaaaaa aa@4141@  is the distance with sign between the Tq circumcenter c T q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgeaYxb9fr=xfr=x fr=BmeaabiqaciaacaGaaeqabaWaaeaaeaaakeaajugibiaadogaju aGdaWgaaqcbasaaKqzadGaamivaSWaaSbaaKGaGeaajugOaiaadgha aKGaGeqaaaWcbeaaaaa@3E9D@  (q = e, ep) and the midpoint of ri,ip, Pi,ip (Figure 2a) (see also Eqs. (19.b)-(19.b) in [29] and Eq. (B.1) in Appendix B), given by the cross product

c i,ip T q =( x c q x i,ip )× ( x i x ip )/ | r i,ip | δ q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqqFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJb WcdaqhaaqcbasaaKqzadGaamyAaiaacYcacaWGPbGaamiCaaqcbasa aKqzadGaamivaSWaaSbaaKGaGeaajugOaiaadghaaKGaGeqaaaaaju gibiabg2da9KqbaoaabmaakeaajugibiaahIhajuaGdaWgaaqcbaua aKqzGdGaam4yaSWaaSbaaKGaGeaajugOaiaadghaaKGaafqaaaWcbe aajugibiabgkHiTiaahIhalmaaBaaajeaqbaqcLbmacaWGPbGaaiil aiaadMgacaWGWbaaleqaaaGccaGLOaGaayzkaaqcLbsacqGHxdaTju aGdaWcgaGcbaqcfa4aaeWaaOqaaKqzGeGaaCiEaKqbaoaaBaaajeai baqcLbmacaWGPbaaleqaaKqzGeGaeyOeI0IaaCiEaKqbaoaaBaaaje aibaqcLbmacaWGPbGaamiCaaWcbeaaaOGaayjkaiaawMcaaaqaaKqb aoaaemaakeaajugibiaahkhajuaGdaWgaaqcbasaaKqzadGaamyAai aacYcacaWGPbGaamiCaaWcbeaaaOGaay5bSlaawIa7aaaajugibiab es7aKLqbaoaaBaaajeaibaqcLbmacaWGXbaaleqaaaaa@7463@  (10),

where xi(ip), xi,ip and x c q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgeaYxb9fr=xfr=x fr=BmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiaahIhaju aGdaWgaaqcbauaaKqzGdGaam4yaSWaaSbaaKGaGeaajugWaiaadgha aKGaafqaaaWcbeaaaaa@3F44@  are the vectors of the coordinates of i (ip), Pi,ip and c T q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgeaYxb9fr=xfr=x fr=BmeaabiqaciaacaGaaeqabaWaaeaaeaaakeaajugibiaadogaju aGdaWgaaqcbauaaKqzGdGaamivaSWaaSbaaKGaGeaajugWaiaadgha aKGaafqaaaWcbeaaaaa@3F1E@ , respectively, δ q = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH0oazkmaaBaaaleaajugWaiaadghaaSqabaqcLbsacqGH 9aqpcaqGGaGaeyOeI0IaaGymaaaa@3EDF@  if the direction of ri,ip is counterclockwise in triangle Tq, δ q = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH0oazkmaaBaaaleaajugWaiaadghaaSqabaqcLbsacqGH 9aqpcaqGGaGaaGymaaaa@3DF2@ otherwise, and "x" is the symbol of the cross product. Eq. (B.2) in Appendix B represents the condition to be satisfied to guarantee the GD condition for side ri,ip.

Figure 2: a) Notations. b) The computational cell (dual finite volume, or Voronoi polygon). In bold black the boundary of the cell.

We construct a dual mesh over Th, where a dual finite volume ei, i =1, ..., N, is associated with node i. It is the Voronoi polygon, defined as in [40], shown in Figure 2b. We call the dual finite volumes, (computational) cells. A node-centered formulation is adopted, with the unknown variables stored in each node i. As motivated at the beginning of section 4.2 and in Appendix C, we concentrate the storage capacity in the nodes in the measure of 1/3 of the area of all the triangles sharing the node.

We will prove that the GD mesh property ensures that 1) the sign of the flux among contiguous cells only depends on the flow rate vector, in the CP and CC hydrostatic steps (in section 4.1), 2) the M-property of the matrix of the systems in the DC hydrostatic step and in the dynamic problem, respectively (in sections 4.2 and 5), and, 3) the sign of the flux among contiguous cells are consistent with the differences of the values of H and qb in the same cells, respectively, in the DC hydrostatic step and in the non-hydrostatic problem (in sections 4.2 and 5). See for example in [41] the definition of a M-matrix.

Let Suppi be the support of cell i, i.e., the set of the NT,i triangles sharing node i. Ai and Nli denote the area and the number of the sides of the boundary of the cell associated to node i, respectively. Ai is computed as

A i = 1 3 T r Sup p i | T r | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGaamyqaK qbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaaIZaaaaKqbaoaaqafake aajuaGdaabdaGcbaqcLbsacaWGubqcfa4aaSbaaKqaGeaajugWaiaa dkhaaSqabaaakiaawEa7caGLiWoaaKqaGeaajugWaiaadsfalmaaBa aajiaibaqcLbmacaWGYbaajiaibeaajugWaiabgIGiolaadofacaWG 1bGaamiCaiaadchalmaaBaaajiaibaqcLbmacaWGPbaajiaibeaaaS qabKqzGeGaeyyeIuoaaaa@5872@  r = 1, ..., NT,i (11).

Hereafter we assume that cell i shares its lth side (l = 1, ..., Nli) with cell ip (Figure 2b). Length with sign dl,i of side l is given by (Figure 2a)

d l,i = q=1,2 c i,ip T l q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGKb qcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaajugi biaayIW7cqGH9aqpcaaMi8UaaGjcVNqbaoaaqafakeaajugibiaado galmaaDaaajeaibaqcLbmacaWGPbGaaiilaiaadMgacaWGWbaajeai baqcLbmacaWGubaddaWgaaqccasaaKqzGcGaamiBaWWaaSbaaKGaGe aajugOaiaadghaaKGaGeqaaaqabaaaaaqcbasaaKqzadGaamyCaiab g2da9iaaigdacaGGSaGaaGOmaaWcbeqcLbsacqGHris5aaaa@5899@  with T l q Sup p i ,  q={ 1 2    T l q ={ T e T ep ,    l=1,...,N l i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaKqaGeaajugWaiaadYgalmaaBaaajiaibaqcLbmacaWG XbaajiaibeaaaSqabaqcLbsacaaMi8UaaGjcVlabgIGiolaayIW7ca aMi8Uaam4uaiaadwhacaWGWbGaamiCaKqbaoaaBaaajeaibaqcLbma caWGPbaaleqaaKqzGeGaaiilaiaabccacaqGGaGaamyCaiaayIW7cq GH9aqpjuaGdaGabaqcLbsaeaqabOqaaKqzGeGaaGymaaGcbaqcLbsa caaIYaaaaOGaay5EaaqcLbsacaqGGaGaeyO0H4Taaeiiaiaadsfaju aGdaWgaaqcbauaaKqzGdGaamiBaSWaaSbaaKGaGeaajugWaiaadgha aKGaafqaaaWcbeaajugibiaayIW7cqGH9aqpjuaGdaGabaqcLbsaea qabOqaaKqzGeGaamivaKqbaoaaBaaajeaibaqcLbmacaWGLbaaleqa aaGcbaqcLbsacaWGubqcfa4aaSbaaKqaGeaajugWaiaadwgacaWGWb aaleqaaaaakiaawUhaaKqzGeGaaiilaiaabccacaqGGaGaaeiiaiaa bccacaWGSbGaeyypa0JaaGymaiaacYcacaGGUaGaaiOlaiaac6caca GGSaGaamOtaiaadYgalmaaBaaajeaibaqcLbmacaWGPbaajeaibeaa aaa@7FC1@  (12),

with c i,ip T l q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJb WcdaqhaaqcbasaaKqzadGaamyAaiaacYcacaWGPbGaamiCaaqcbasa aKqzadGaamivaSWaaSbaaKGaGeaajugWaiaadYgalmaaBaaajiaiba qcLbmacaWGXbaajiaibeaaaeqaaaaaaaa@43BE@  defined in Eq. (10) and Eq. (B.1) in Appendix B. According to the definition of c i,ip T l q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJb WcdaqhaaqcbasaaKqzadGaamyAaiaacYcacaWGPbGaamiCaaqcbasa aKqzadGaamivaSWaaSbaaKGaGeaajugWaiaadYgalmaaBaaajiaiba qcLbmacaWGXbaajiaibeaaaeqaaaaaaaa@43BE@ , side l is orthogonal to ri,ip. From Eqs. (10), (12) and (B.1) in Appendix B, dl,i could be either positive, either negative. In a GD mesh, dl,i is always positive (see Eq. (10) and Appendix B). The boundary of the cell i is defined as Γ i = l=1,N l i d l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqqHto WrjuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajugibiabg2da9Kqb aoaaqabakeaajugibiaadsgajuaGdaWgaaqcbasaaKqzadGaamiBai aacYcacaWGPbaaleqaaaqaaKqzGeGaamiBaiabg2da9iaaigdacaGG SaGaamOtaiaadYgajuaGdaWgaaqccasaaKqzadGaamyAaaadbeaaaS qabKqzGeGaeyyeIuoaaaa@4E19@ , always positive in a GD mesh.

The Hydrostatic Problem

Eqs. (8)-(9) are integrated in space (see also Eqs. (A.3)-(A.5) in Appendix A), and, using the Green’s theorem, we get the integral formulation

h i t A i + l=1,N l i F l,i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabiqaciaacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaa GcbaqcLbsacqGHciITcaWGObqcfa4aaSbaaKqaGeaajugWaiaadMga aSqabaaakeaajugibiabgkGi2kaadshaaaGaamyqaKqbaoaaBaaaje aibaqcLbmacaWGPbaaleqaaKqzGeGaey4kaSscfa4aaabuaOqaaKqz GeGaamOraKqbaoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaS qabaaajeaibaqcLbmacaWGSbGaeyypa0JaaGymaiaacYcacaWGobGa amiBaSWaaSbaaKGaGeaajugWaiaadMgaaKGaGeqaaaWcbeqcLbsacq GHris5aiabg2da9iaaicdaaaa@5926@  i = 1, …, N                                               (13),

q s,i t A i + l=1,N l i M l,i s + R i s =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabiqaciaacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaa GcbaqcLbsacqGHciITcaWGXbqcfa4aaSbaaKqaGeaajugWaiaadoha caGGSaGaamyAaaWcbeaaaOqaaKqzGeGaeyOaIyRaamiDaaaacaWGbb qcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaqcLbsacqGHRaWkjuaG daaeqbGcbaqcLbsacaaMc8UaamytaSWaa0baaKqaGeaajugWaiaadY gacaGGSaGaamyAaaqcbasaaKqzadGaam4CaaaaaKqaGeaajugWaiaa dYgacqGH9aqpcaaIXaGaaiilaiaad6eacaWGSbWcdaWgaaqccasaaK qzadGaamyAaaqccasabaaaleqajugibiabggHiLdGaey4kaSIaamOu aSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWaiaadohaaaqcLb sacqGH9aqpcaaIWaaaaa@6537@  s = x, y                                      (14.a),

s={ x y    q s ={ Uh Vh ,    M l,i s ={ M l,i x M l,i y ,    R i s ={ R i x = A i g( h i H i k x + ( Uh ) i st ) R i y = A i g( h i H i k y + ( Vh ) i st )  st= n i 2 ( Uh ) i 2 + ( Vh ) i 2 h i 7/3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb GaaGjcVlabg2da9KqbaoaaceaakeaajugibuaabeqaceaaaOqaaKqz GeGaamiEaaGcbaqcLbsacaWG5baaaaGccaGL7baajugibiaabccacq GHshI3caqGGaGaamyCaKqbaoaaBaaajeaibaqcLbmacaWGZbaaleqa aKqzGeGaaGjcVlabg2da9iaayIW7juaGdaGabaGcbaqcLbsafaqabe GabaaakeaajugibiaadwfacaWGObaakeaajugibiaadAfacaWGObaa aaGccaGL7baajugibiaacYcacaqGGaGaaeiiaiaabccacaWGnbWcda qhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbmacaWG ZbaaaKqzGeGaaGjcVlabg2da9iaayIW7juaGdaGabaGcbaqcLbsafa qabeGabaaakeaajugibiaad2ealmaaDaaajeaibaqcLbmacaWGSbGa aiilaiaadMgaaKqaGeaajugWaiaadIhaaaaakeaajugibiaad2ealm aaDaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaKqaGeaajugWaiaa dMhaaaaaaaGccaGL7baajugibiaacYcacaqGGaGaaeiiaiaabccaca WGsbWcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaam4Caaaa jugibiaayIW7cqGH9aqpjuaGdaGabaGcbaqcLbsafaqabeGabaaake aajugibiaadkfalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbma caWG4baaaKqzGeGaaGjcVlabg2da9iaayIW7caWGbbqcfa4aaSbaaK qaGeaajugWaiaadMgaaSqabaqcLbsacaWGNbqcfa4aaeWaaOqaaKqz GeGaamiAaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqzGeGaaG jcVNqbaoaalaaakeaajugibiabgkGi2kaadIealmaaDaaajeaibaqc LbmacaWGPbaajeaibaqcLbmacaWGRbaaaaGcbaqcLbsacqGHciITca WG4baaaiabgUcaRKqbaoaabmaakeaajugibiaadwfacaWGObaakiaa wIcacaGLPaaajuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajugibi aayIW7caaMi8Uaam4CaiaadshaaOGaayjkaiaawMcaaaqaaKqzGeGa amOuaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWaiaadMhaaa qcLbsacaaMi8Uaeyypa0JaaGjcVlaadgeajuaGdaWgaaqcbasaaKqz adGaamyAaaWcbeaajugibiaadEgajuaGdaqadaGcbaqcLbsacaWGOb qcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaqcLbsacaaMi8Ecfa4a aSaaaOqaaKqzGeGaeyOaIyRaamisaSWaa0baaKqaGeaajugWaiaadM gaaKqaGeaajugWaiaadUgaaaaakeaajugibiabgkGi2kaadMhaaaGa ey4kaSscfa4aaeWaaOqaaKqzGeGaamOvaiaadIgaaOGaayjkaiaawM caaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqzGeGaam4Caiaa dshaaOGaayjkaiaawMcaaaaaaiaawUhaaKqzGeGaaeiiaiaadohaca WG0bGaeyypa0tcfa4aaSaaaOqaaKqzGeGaamOBaSWaa0baaKqaGeaa jugWaiaadMgaaKqaGeaajugWaiaaikdaaaqcLbsacaaMi8Ecfa4aaO aaaOqaaKqbaoaabmaakeaajugibiaadwfacaWGObaakiaawIcacaGL PaaalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaaIYaaaaK qzGeGaaGjcVlabgUcaRKqbaoaabmaakeaajugibiaadAfacaWGObaa kiaawIcacaGLPaaalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLb macaaIYaaaaaWcbeaaaOqaaKqzGeGaamiAaSWaa0baaKqaGeaajugW aiaadMgaaKqaGeaajugWaiaaiEdacaGGVaGaaG4maaaaaaaaaa@0E33@             (14.b),

where sub index i marks the variables in cell i, Fl,i and M l,i x( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgeaYxb9fr=xfr=x fr=BmeaabiqaciaacaGaaeqabaWaaeaaeaaakeaajugibiaad2ealm aaDaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaKqaGeaajugWaiaa dIhalmaabmaajeaibaqcLbmacaWG5baajeaicaGLOaGaayzkaaaaaa aa@4393@ , further specified, are the flux and the x(y) component of the momentum flux across side l (l = 1, ..., Nli) of cell i and R i x( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgeaYxb9fr=xfr=x fr=BmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiaadkfalm aaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWG4bWcdaqadaqc basaaKqzadGaamyEaaqcbaIaayjkaiaawMcaaaaaaaa@41F5@  is the source term. The Manning coefficient in cell i, ni is computed as the mean value of the friction coefficients in the triangles sharing node i. The spatial gradients of the water level are kept constant in the prediction step of the hydrostatic problem (see Eq (14,b)), as motivated in [31,32].

In accordance to the formulation in Eqs. (A.3)-(A.5) in Appendix A, the linearized differential formulation of the DC problem is

h t + Uh x + Vh y = ( Uh ¯ ) x + ( Vh ¯ ) y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabiqaciaacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaa GcbaqcLbsacqGHciITcaWGObaakeaajugibiabgkGi2kaadshaaaGa ey4kaSscfa4aaSaaaOqaaKqzGeGaeyOaIyRaamyvaiaadIgaaOqaaK qzGeGaeyOaIyRaamiEaaaacqGHRaWkjuaGdaWcaaGcbaqcLbsacqGH ciITcaWGwbGaamiAaaGcbaqcLbsacqGHciITcaWG5baaaiabg2da9K qbaoaalaaakeaajugibiabgkGi2MqbaoaabmaakeaajuaGdaqdaaGc baqcLbsacaWGvbGaamiAaaaaaOGaayjkaiaawMcaaaqaaKqzGeGaey OaIyRaamiEaaaacqGHRaWkjuaGdaWcaaGcbaqcLbsacqGHciITjuaG daqadaGcbaqcfa4aa0aaaOqaaKqzGeGaamOvaiaadIgaaaaakiaawI cacaGLPaaaaeaajugibiabgkGi2kaadMhaaaaaaa@64BD@  (15),

q s t +g h ¯ H s +g n 2 ( q s ) ( ( Uh ) 2 + ( Vh ) 2 h 7/3 ) ¯ =g h ¯ H k s +g n 2 ( q ¯ s ) ( ( Uh ) 2 + ( Vh ) 2 h 7/3 ) ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabiqaciaacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaa GcbaqcLbsacqGHciITcaWGXbqcfa4aaSbaaKqaGeaajugWaiaadoha aSqabaaakeaajugibiabgkGi2kaadshaaaGaey4kaSIaam4zaiqadI gagaqeaKqbaoaalaaakeaajugibiabgkGi2kaadIeaaOqaaKqzGeGa eyOaIyRaam4CaaaacqGHRaWkcaWGNbGaaGPaVlaad6gajuaGdaahaa WcbeqcbasaaKqzadGaaGOmaaaajuaGdaqadaGcbaqcLbsacaWGXbqc fa4aaSbaaKqaGeaajugWaiaadohaaSqabaaakiaawIcacaGLPaaaju aGdaqdaaGcbaqcfa4aaeWaaOqaaKqbaoaalaaakeaajuaGdaGcaaGc baqcfa4aaeWaaOqaaKqzGeGaamyvaiaadIgaaOGaayjkaiaawMcaaK qbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSscfa4a aeWaaOqaaKqzGeGaamOvaiaadIgaaOGaayjkaiaawMcaaKqbaoaaCa aaleqajeaibaqcLbmacaaIYaaaaaWcbeaaaOqaaKqzGeGaamiAaKqb aoaaCaaaleqajeaibaqcLbmacaaI3aGaai4laiaaiodaaaaaaaGcca GLOaGaayzkaaaaaKqzGeGaeyypa0Jaam4zaiqadIgagaqeaKqbaoaa laaakeaajugibiabgkGi2kaadIeajuaGdaahaaWcbeqcbasaaKqzad Gaam4AaaaaaOqaaKqzGeGaeyOaIyRaam4CaaaacqGHRaWkcaWGNbGa aGPaVlaad6gajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajuaGda qadaGcbaqcLbsaceWGXbGbaebajuaGdaWgaaWcbaqcLbsacaWGZbaa leqaaaGccaGLOaGaayzkaaqcfa4aa0aaaOqaaKqbaoaabmaakeaaju aGdaWcaaGcbaqcfa4aaOaaaOqaaKqbaoaabmaakeaajugibiaadwfa caWGObaakiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaKqzadGaaG OmaaaajugibiabgUcaRKqbaoaabmaakeaajugibiaadAfacaWGObaa kiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaS qabaaakeaajugibiaadIgajuaGdaahaaWcbeqcbasaaKqzadGaaG4n aiaac+cacaaIZaaaaaaaaOGaayjkaiaawMcaaaaaaaa@A526@  (16),

and the symbol ( * ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceGGQa Gbaebaaaa@374A@ ) marks the mean-in-time value of the variable (*). The initial states of the CC and DC steps are the solutions of the CP and CC steps, respectively. In Eqs. (16) we assume negligible the difference between the sum of the convective terms and the mean-in-time values of the same quantities obtained after the (CP + CC) steps [31,32]. Symbols (*)k, (*)k+1/3, (*)k+2/3, ( ( * ˜ ) k+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GabiOkayaaiaGaaiykaOWaaWbaaSqabeaacaWGRbGaey4kaSIaaGym aaaaaaa@3B5E@  mark the values of the variable (*) at the beginning of the time iteration, at the end of the convective prediction, convective correction and diffusive correction steps, respectively.

In the prediction steps, we assume, inside each computational cell, piecewise constant values of h, Uh, Vh. The correction system (15)-(16) is solved applying a technique similar to the linear (P1) conforming FE Galerkin scheme (see section 4.2). The procedure for the solution of the hydrostatic problem has been proposed in [32], where we adopted a cell-centered formulation with storage capacity concentrated in the mesh triangles. The node-centered formulation with respect to the Voronoi cells, presented in this paper, leads to a different spatial discretization of the governing equations. In the following sections 4.1 and 4.2, we only report the steps of the numerical formulation of the hydrostatic problem involved in the new spatial discretization. In test 3 (section 8.3) we compare the efficiency and accuracy of the two formulations using an analytical reference solution.

The prediction problem

All the computational cells, at the beginning of each time iteration, are ordered according to the value of the approximated potential ϕ i k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaaaa @3D06@ , known inside each cell, and computed according to the procedure proposed in [31,32]. This procedure, at the beginning of the time step, minimizes the differences [31,32]. From Eq. (12), we set the flux FLl,i through side l as

F L l.i =( n ^ i,ip q i ) d l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb GaamitaKqbaoaaBaaajeaibaqcLbmacaWGSbGaaiOlaiaadMgaaSqa baqcLbsacqGH9aqpjuaGdaqadaGcbaqcLbsaceWHUbGbaKaajuaGda WgaaqcbasaaKqzadGaamyAaiaacYcacaWGPbGaamiCaaWcbeaajugi biabgwSixlaahghajuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaaaO GaayjkaiaawMcaaKqzGeGaamizaKqbaoaaBaaajeaibaqcLbmacaWG SbGaaiilaiaadMgaaSqabaaaaa@5497@                               (17),

where n ^ i,ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiqah6 gagaqcaKqbaoaaBaaajeaibaqcLbmacaWGPbGaaiilaiaadMgacaWG Wbaaleqaaaaa@3D35@  is the unit vector parallel to side ri,ip (i.e., orthogonal to side l) positive outwards from i, i.e., oriented from i to ip, qi is the specific horizontal flow rate vector in cell i, qi = ((Uh)i, (Vh)i)T. and symbol "" MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqqFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaiOiaiabgw Sixlaackcaaaa@394B@  marks the dot product. Eq. (17) becomes

F L l,i =( ( Uh ) i ( x ip x i )+ ( Vh ) i ( y ip y i ) ) d l,i / | r i,ip | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb GaamitaKqbaoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqa baqcLbsacqGH9aqpjuaGdaqadaGcbaqcfa4aaeWaaOqaaKqzGeGaam yvaiaadIgaaOGaayjkaiaawMcaaKqbaoaaBaaajeaibaqcLbmacaWG PbaaleqaaKqbaoaabmaakeaajugibiaadIhajuaGdaWgaaqcbasaaK qzadGaamyAaiaadchaaSqabaqcLbsacqGHsislcaWG4bqcfa4aaSba aKqaGeaajugWaiaadMgaaSqabaaakiaawIcacaGLPaaajugibiabgU caRKqbaoaabmaakeaajugibiaadAfacaWGObaakiaawIcacaGLPaaa juaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajuaGdaqadaGcbaqcLb sacaWG5bqcfa4aaSbaaKqaGeaajugWaiaadMgacaWGWbaaleqaaKqz GeGaeyOeI0IaamyEaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaa GccaGLOaGaayzkaaaacaGLOaGaayzkaaqcfa4aaSGbaOqaaKqzGeGa amizaKqbaoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqaba aakeaajuaGdaabdaGcbaqcLbsacaWHYbqcfa4aaSbaaKqaGeaajugW aiaadMgacaGGSaGaamyAaiaadchaaSqabaaakiaawEa7caGLiWoaaa aaaa@7CE8@  (18).

If the GD property holds for side ri,ip, dl,i is positive (see Eq. (10) and Appendix B), and the sign of FLl,i only depends on the dot product ( n ^ i,ip q i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGabCOBayaajaqcfa4aaSbaaKqaGeaajugWaiaadMgacaGG SaGaamyAaiaadchaaSqabaqcLbsacqGHflY1caWHXbqcfa4aaSbaaK qaGeaajugWaiaadMgaaSqabaaakiaawIcacaGLPaaaaaa@461B@ , and a leaving (incoming) flux from (in) cell i is positive (negative). This condition is not fulfilled if the GD property is not satisfied, since dl,i is negative. The fluxes and momentum fluxes between cells i and ip in Eqs. (13)-(14) are defined as

F l,i =F L l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaajugi biabg2da9iaaysW7caWGgbGaamitaKqbaoaaBaaajeaibaqcLbmaca WGSbGaaiilaiaadMgaaSqabaaaaa@454E@  if ( F L l,i >0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb GaamitaKqbaoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqa baqcLbsacqGH+aGpcaaIWaaaaa@3F11@  and F L l,i >F L m,ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb GaamitaKqbaoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqa baqcLbsacqGH+aGpcaWGgbGaamitaKqbaoaaBaaajeaibaqcLbmaca WGTbGaaiilaiaadMgacaWGWbaaleqaaaaa@458A@ ) F l,i =F L m,ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaajugi biabg2da9iabgkHiTiaadAeacaWGmbqcfa4aaSbaaKqaGeaajugWai aad2gacaGGSaGaamyAaiaadchaaSqabaaaaa@45A4@  otherwise (19),

M l,i s = F l,i q s,i / h i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb WcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbma caWGZbaaaKqzGeGaaGzaVlabg2da9iaaxcW7caaMb8UaamOraSWaaS baaKqaGeaajugWaiaadYgacaGGSaGaamyAaaqcbasabaqcfa4aaSGb aOqaaKqzGeGaamyCaKqbaoaaBaaajeaibaqcLbmacaWGZbGaaiilai aadMgaaSqabaaakeaajugibiaadIgajuaGdaWgaaqcbasaaKqzadGa amyAaaWcbeaaaaaaaa@5453@  if F l,i =F L l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaajugi biaaygW7caaMc8Uaeyypa0JaaGzaVlaaykW7caWGgbGaamitaKqbao aaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqabaaaaa@49EB@  , M l,i s = F l,i q s,ip / h ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb WcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbma caWGZbaaaKqzGeGaaGzaVlabg2da9iaaygW7caaMe8UaamOraKqbao aaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqabaqcfa4aaSGb aOqaaKqzGeGaamyCaKqbaoaaBaaajeaibaqcLbmacaWGZbGaaiilai aadMgacaWGWbaaleqaaaGcbaqcLbsacaWGObqcfa4aaSbaaKqaGeaa jugWaiaadMgacaWGWbaaleqaaaaaaaa@56A6@  otherwise                         (20),

where m is the side of cell ip shared with cell i. Eqs. (19)-(20) guarantee the continuity of the flux and the momentum flux at each internal cell side, so that F l,i = F m,ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaajugi biabg2da9iabgkHiTiaadAeajuaGdaWgaaqcbasaaKqzadGaamyBai aacYcacaWGPbGaamiCaaWcbeaaaaa@44D3@  and M l,i x( y ) = M m,ip x( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb WcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbma caWG4bWcdaqadaqcbasaaKqzadGaamyEaaqcbaIaayjkaiaawMcaaa aajugibiabg2da9iaaysW7cqGHsislcaWGnbWcdaqhaaqcbasaaKqz adGaamyBaiaacYcacaWGPbGaamiCaaqcbasaaKqzadGaamiEaSWaae WaaKqaGeaajugWaiaadMhaaKqaGiaawIcacaGLPaaaaaaaaa@5226@  . For an external boundary side, the condition Fl,i = Fll,i holds for an outward oriented (positive) flux.

Thanks to the consecutive solution of the cells performed in the MAST procedure, the system of PDEs (13)-(14) for each cell i are written as a system of Ordinary Differential Equations (ODEs) (more details in [31-33]). After the cells ordering, in the CP step we proceed from the cell with the highest to the cell with the lowest value of ϕ k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2aaWbaaSqabeaacaWGRbaaaaaa@394B@ , and solve a ODEs system for each cell in the interval [ 0, Δt ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaqcLb saqaaaaaaaaaWdbiaaicdacaGGSaGaaeiiaiabfs5aejaadshaaOWd aiaawUfacaGLDbaaaaa@3D6C@  (from time level tk to time level tk+1/3and Δt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqqHuoarcaWG0baaaa@3954@  is the size of the time step). After that, in the CC step, we proceed from the cell with the lowest to the cell with the highest value of ϕ k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2aaWbaaSqabeaacaWGRbaaaaaa@394B@ , solving an ODEs system in each cell in the interval [ 0, Δt ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaqcLb saqaaaaaaaaaWdbiaaicdacaGGSaGaaeiiaiabfs5aejaadshaaOWd aiaawUfacaGLDbaaaaa@3D6C@  (from time level tk+1/3 to time level tk+2/3).

A 5th order Runge-Kutta method with adaptive stepsize control [42] is applied for the solution of the ODEs system. The ODEs system for the CP step is

d h i dt A i + 1 Δt Δt l=1,N l i δ l,i F l,i p,out ( t )dt = l=1,N l i ( 1 δ l,i ) F ¯ l,i p,in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadIgajuaGdaWgaaqcbasaaKqzadGaamyAaaWc beaaaOqaaKqzGeGaamizaiaadshaaaGaamyqaKqbaoaaBaaajeaiba qcLbmacaWGPbaaleqaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGa aGymaaGcbaqcLbsacqqHuoarcaWG0baaaKqbaoaapefakeaajuaGda aeqbGcbaqcLbsacqaH0oazjuaGdaWgaaqcbasaaKqzadGaamiBaiaa cYcacaWGPbaaleqaaKqzGeGaamOraKqbaoaaDaaajeaibaqcLbmaca WGSbGaaiilaiaadMgaaKqaGeaajugWaiaadchacaGGSaGaam4Baiaa dwhacaWG0baaaKqbaoaabmaakeaajugibiaadshaaOGaayjkaiaawM caaKqzGeGaamizaiaadshaaKqaGeaajugWaiaadYgacqGH9aqpcaaI XaGaaiilaiaad6eacaWGSbWcdaWgaaqccasaaKqzadGaamyAaaqcca sabaaaleqajugibiabggHiLdaajeaibaqcLbmacqqHuoarcaWG0baa leqajugibiabgUIiYdGaeyypa0tcfa4aaabuaOqaaKqbaoaabmaake aajugibiaaigdacqGHsislcqaH0oazjuaGdaWgaaqcbasaaKqzadGa amiBaiaacYcacaWGPbaaleqaaaGccaGLOaGaayzkaaqcLbsaceWGgb GbaebalmaaDaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaKqaGeaa jugWaiaadchacaGGSaGaamyAaiaad6gaaaaajeaibaqcLbmacaWGSb Gaeyypa0JaaGymaiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugW aiaadMgaaKGaGeqaaaWcbeqcLbsacqGHris5aaaa@974A@  (21),

d q s,i ( t ) dt A i + 1 Δt Δt R i p,s ( t )dt + 1 Δt Δt l=1,N l i δ l,i M l,i p,s,out ( t )dt = l=1,N l i ( 1 δ l,i ) M ¯ l,i p,s,in ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadghajuaGdaWgaaqcbasaaKqzadGaam4Caiaa cYcacaWGPbaaleqaaKqbaoaabmaakeaajugibiaadshaaOGaayjkai aawMcaaaqaaKqzGeGaamizaiaadshaaaGaamyqaKqbaoaaBaaajeai baqcLbmacaWGPbaaleqaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGe GaaGymaaqcaasaaKqzadGaeuiLdqKaamiDaaaajuaGdaWdrbGcbaqc LbsacaWGsbWcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaam iCaiaacYcacaWGZbaaaKqbaoaabmaakeaajugibiaadshaaOGaayjk aiaawMcaaKqzGeGaamizaiaadshaaKqaGeaajugWaiabfs5aejaads haaSqabKqzGeGaey4kIipacqGHRaWkjuaGdaWcaaGcbaqcLbsacaaI XaaajaaibaqcLbmacqqHuoarcaWG0baaaKqbaoaapefakeaajuaGda aeqbGcbaqcLbsacqaH0oazjuaGdaWgaaqcbasaaKqzadGaamiBaiaa cYcacaWGPbaaleqaaKqzGeGaamytaSWaa0baaKqaGeaajugWaiaadY gacaGGSaGaamyAaaqcbasaaKqzadGaamiCaiaacYcacaWGZbGaaiil aiaad+gacaWG1bGaamiDaaaajuaGdaqadaGcbaqcLbsacaWG0baaki aawIcacaGLPaaajugibiaadsgacaWG0baajeaibaqcLbmacaWGSbGa eyypa0JaaGymaiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugWai aadMgaaKGaGeqaaaWcbeqcLbsacqGHris5aaqcbasaaKqzadGaeuiL dqKaamiDaaWcbeqcLbsacqGHRiI8aiabg2da9Kqbaoaaqafakeaaju aGdaqadaGcbaqcLbsacaaIXaGaeyOeI0IaeqiTdqwcfa4aaSbaaKqa GeaajugWaiaadYgacaGGSaGaamyAaaWcbeaaaOGaayjkaiaawMcaaK qzGeGabmytayaaraWcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWG PbaajeaibaqcLbmacaWGWbGaaiilaiaadohacaGGSaGaamyAaiaad6 gaaaqcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaaajeai baqcLbmacaWGSbGaeyypa0JaaGymaiaacYcacaWGobGaamiBaSWaaS baaKGaGeaajugWaiaadMgaaKGaGeqaaaWcbeqcLbsacqGHris5aaaa @BF87@  s = x, y (22),

with R i x( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsb WcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaamiEaSWaaeWa aKqaGeaajugWaiaadMhaaKqaGiaawIcacaGLPaaaaaaaaa@4036@  defined in Eqs. (14,b), δ li =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKnaaBa aaleaacaWGSbGaamyAaaqabaGccqGH9aqpcaaIXaaaaa@3BC1@  if flux across side l is oriented outward i , 0 otherwise and
F l,i p,out =max( 0, F l,i ), M l,i p,s,out = F l,i p,out q s,i / h i ,s=x, y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb WcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbma caWGWbGaaiilaiaad+gacaWG1bGaamiDaaaajugibiabg2da9iGac2 gacaGGHbGaaiiEaKqbaoaabmaakeaajugibiaaicdacaGGSaGaamOr aSWaaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaqcbasabaaaki aawIcacaGLPaaajuaGcaGGSaGaaGPaVlaaykW7caaMc8EcLbsacaWG nbWcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLb macaWGWbGaaiilaiaadohacaGGSaGaam4BaiaadwhacaWG0baaaKqz GeGaeyypa0JaamOraSWaa0baaKqaGeaajugWaiaadYgacaGGSaGaam yAaaqcbasaaKqzadGaamiCaiaacYcacaWGVbGaamyDaiaadshaaaqc fa4aaSGbaOqaaKqzGeGaamyCaKqbaoaaBaaajeaibaqcLbmacaWGZb GaaiilaiaadMgaaSqabaaakeaajugibiaadIgajuaGdaWgaaqcbasa aKqzadGaamyAaaWcbeaaaaqcLbsacaGGSaGaaGzbVlaadohacqGH9a qpcaWG4bGaaiilaiaabccacaWG5baaaa@818D@ , if ϕ i k ϕ ip k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaKqz GeGaeyyzImRaeqy1dy2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaK qaGeaajugWaiaadUgaaaaaaa@46D3@  (23.a),

F l,i p,in =min( 0, F l,i ), M l,i p,x,in = F l,i p,in q s,ip / h ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb WcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbma caWGWbGaaiilaiaadMgacaWGUbaaaKqzGeGaeyypa0JaciyBaiaacM gacaGGUbqcfa4aaeWaaOqaaKqzGeGaaGimaiaacYcacaWGgbqcfa4a aSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaaaOGaayjkai aawMcaaKqzGeGaaiilaiaaykW7caaMc8UaaGPaVlaad2ealmaaDaaa jeaibaqcLbmacaWGSbGaaiilaiaadMgaaKqaGeaajugWaiaadchaca GGSaGaamiEaiaacYcacaWGPbGaamOBaaaajugibiabg2da9iaadAea lmaaDaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaKqaGeaajugWai aadchacaGGSaGaamyAaiaad6gaaaqcfa4aaSGbaOqaaKqzGeGaamyC aKqbaoaaBaaajeaibaqcLbmacaWGZbGaaiilaiaadMgacaWGWbaale qaaaGcbaqcLbsacaWGObqcfa4aaSbaaKqaGeaajugWaiaadMgacaWG Wbaaleqaaaaaaaa@7825@ , if ϕ i k < ϕ ip k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaKqz GeGaeyipaWJaeqy1dy2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaK qaGeaajugWaiaadUgaaaaaaa@4611@                                (23.b).

The right hand side (r.h.s.) of Eqs. (21)-(22) are the mean-in-time values of the incoming flux and x/y momentum fluxes, respectively, known after the solution of the linked ip cells with higher value of ϕ k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqy1dy McdaahaaWcbeqcbasaaKqzadGaam4Aaaaaaaa@3B1C@ , solved before i [31,32]. The gradients of H in the Rx(y) term are computed as proposed in 4.3.

Once the ODEs (21)-(22) are solved for cell i, the mean-in-time value of the total flux F ¯ i out MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaraWcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaam4Baiaa dwhacaWG0baaaaaa@3D26@ leaving from i, is given by the local mass balance [32],

F ¯ i out = F ¯ i in ( h i k+1/3 h i k )/ Δt A i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaamOraaaalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqc LbmacaWGVbGaamyDaiaadshaaaqcLbsacqGH9aqpjuaGdaqdaaGcba qcLbsacaWGgbaaaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugW aiaadMgacaWGUbaaaKqzGeGaeyOeI0scfa4aaSGbaOqaaKqbaoaabm aakeaajugibiaadIgalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqc LbmacaWGRbGaey4kaSIaaGymaiaac+cacaaIZaaaaKqzGeGaeyOeI0 IaamiAaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWaiaadUga aaaakiaawIcacaGLPaaaaeaajugibiabfs5aejaadshaaaGaamyqaK qbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaaaa@635E@  (24),

where F ¯ i in = l=1,N l i ( 1 δ l,i ) F ¯ l,i p,in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaamOraaaalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqc LbmacaWGPbGaamOBaaaajugibiabg2da9KqbaoaaqafakeaajuaGda qadaGcbaqcLbsacaaIXaGaeyOeI0IaeqiTdq2cdaWgaaqcbasaaKqz adGaamiBaiaacYcacaWGPbaajeaibeaaaOGaayjkaiaawMcaaKqzGe GabmOrayaaraWcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaa jeaibaqcLbmacaWGWbGaaiilaiaadMgacaWGUbaaaaqcbasaaKqzad GaamiBaiabg2da9iaaigdacaGGSaGaamOtaiaadYgalmaaBaaajiai baqcLbmacaWGPbaajiaibeaaaSqabKqzGeGaeyyeIuoaaaa@6039@  (i.e., the r.h.s. of Eq. (21)). Following [32], we compute the mean-in-time value of the flux F ¯ l,i out MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaraWcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqc LbmacaWGVbGaamyDaiaadshaaaaaaa@3EC7@ leaving from side l of i to the neighboring cell ip with ϕ i k ϕ ip k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaKqz GeGaeyyzImRaeqy1dy2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaK qaGeaajugWaiaadUgaaaaaaa@46D3@ , by partitioning F ¯ i out MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaraWcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaam4Baiaa dwhacaWG0baaaaaa@3D26@  on the base of the ratio between the flux F l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGaamOraK qbaoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqabaaaaa@3AFE@  and the sum of all the leaving fluxes to the neighboring cells at the end of the time step,

F ¯ l,i out = F ¯ i out ( F l,i p, out ) k+1/3 l=1,N l i δ l,i ( F l,i p, out ) k+1/3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaraqcfa4aa0baaKqaGeaajugWaiaadYgacaGGSaGaamyAaaqcbasa aKqzadGaam4BaiaadwhacaWG0baaaKqzGeGaeyypa0JabmOrayaara qcfa4aa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWaiaad+gacaWG 1bGaamiDaaaajuaGdaqadaGcbaqcLbsacaWGgbqcfa4aa0baaKqaGe aajugWaiaadYgacaGGSaGaamyAaaqcbasaaKqzadGaamiCaiaacYca caqGGaGaam4BaiaadwhacaWG0baaaaGccaGLOaGaayzkaaqcfa4aaW baaSqabKqaGeaajugWaiaadUgacqGHRaWkcaaIXaGaai4laiaaioda aaGcdaWcdaqaaaqaaaaajugibiabggHiLNqbaoaaBaaaleaajugWai aadYgacqGH9aqpcaaIXaGaaiilaiaad6eacaWGSbWcdaWgaaadbaqc LbmacaWGPbaameqaaaWcbeaajugibiabes7aKLqbaoaaBaaajeaiba qcLbmacaWGSbGaaiilaiaadMgaaSqabaqcfa4aaeWaaOqaaKqzGeGa amOraKqbaoaaDaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaKqaGe aajugWaiaadchacaGGSaGaaeiiaiaad+gacaWG1bGaamiDaaaaaOGa ayjkaiaawMcaaKqbaoaaCaaaleqajeaibaqcLbmacaWGRbGaey4kaS IaaGymaiaac+cacaaIZaaaaaaa@84FB@  (25.a),

and we estimate in a similar way the mean-in-time values of the leaving momentum flux M ¯ l,i x( y ),out MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabiqaciaacaGaaeqabaWaaeaaeaaakeaajugibiqad2 eagaqeaSWaa0baaKqaGeaajugWaiaadYgacaGGSaGaamyAaaqcbasa aKqzadGaamiEaSWaaeWaaKqaGeaajugWaiaadMhaaKqaGiaawIcaca GLPaaajugWaiaacYcacaWGVbGaamyDaiaadshaaaaaaa@4715@ . Finally, we set:

F ¯ m,ip in = F ¯ l,i out MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FasPYRqj0=yi0dg9Gr=dir=f0=yqaiVgFr0xfr=x fr=xgbpdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGgb GbaebalmaaDaaajeaibaqcLbmacaWGTbGaaiilaiaadMgacaWGWbaa jeaibaqcLbmacaWGPbGaamOBaaaajugibiabg2da9iqadAeagaqeaS Waa0baaKqaGeaajugWaiaadYgacaGGSaGaamyAaaqcbasaaKqzadGa am4BaiaadwhacaWG0baaaaaa@4CF6@   M ¯ m,ip s,in = M ¯ l,i s,out           s=x, y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FasPYRqj0=yi0dg9Gr=dir=f0=yqaiVgFr0xfr=x fr=xgbpdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGnb GbaebalmaaDaaajeaibaqcLbmacaWGTbGaaiilaiaadMgacaWGWbaa jeaibaqcLbmacaWGZbGaaiilaiaadMgacaWGUbaaaKqzGeGaeyypa0 JabmytayaaraWcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaa jeaibaqcLbmacaWGZbGaaiilaiaad+gacaWG1bGaamiDaaaacaqGGa qcLbsacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaWGZbGaeyypa0JaamiEaiaacYcacaqGGaGaamyEaa aa@5C8D@ (25.b),

for all the neighboring ip cells with ϕ i k ϕ ip k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaKqz GeGaeyyzImRaeqy1dy2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaK qaGeaajugWaiaadUgaaaaaaa@46D3@  and we proceed to solve system (21)-(22) for the ip cells among the unsolved ones. A similar algorithm is applied to solve the convective correction step, whose ODEs system for the generic cell i is

d h i dt A i + 1 Δt Δt l=1,N l i δ l,i F l,i c,out ( t )dt = l=1,N l i ( 1 δ l,i ) F ¯ l,i c,in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadIgajuaGdaWgaaWcbaqcLbsacaWGPbaaleqa aaGcbaqcLbsacaWGKbGaamiDaaaacaWGbbqcfa4aaSbaaKqaGeaaju gWaiaadMgaaSqabaqcLbsacqGHRaWkjuaGdaWcaaGcbaqcLbsacaaI Xaaakeaajugibiabfs5aejaadshaaaqcfa4aa8quaOqaaKqbaoaaqa fakeaajugibiabes7aKLqbaoaaBaaajeaibaqcLbmacaWGSbGaaiil aiaadMgaaSqabaqcLbsacaWGgbWcdaqhaaqcbasaaKqzadGaamiBai aacYcacaWGPbaajeaibaqcLbmacaWGJbGaaiilaiaad+gacaWG1bGa amiDaaaajuaGdaqadaGcbaqcLbsacaWG0baakiaawIcacaGLPaaaju gibiaadsgacaWG0baajeaibaqcLbmacaWGSbGaeyypa0JaaGymaiaa cYcacaWGobGaamiBaSWaaSbaaKGaGeaajugWaiaadMgaaKGaGeqaaa WcbeqcLbsacqGHris5aaWcbaqcLbsacqqHuoarcaWG0baaleqajugi biabgUIiYdGaeyypa0tcfa4aaabuaOqaaKqbaoaabmaakeaajugibi aaigdacqGHsislcqaH0oazjuaGdaWgaaqcbasaaKqzadGaamiBaiaa cYcacaWGPbaaleqaaaGccaGLOaGaayzkaaqcLbsaceWGgbGbaebalm aaDaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaKqaGeaajugWaiaa dogacaGGSaGaamyAaiaad6gaaaaajeaibaqcLbmacaWGSbGaeyypa0 JaaGymaiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugWaiaadMga aKGaGeqaaaWcbeqcLbsacqGHris5aaaa@9531@  (26),

d q s,i ( t ) dt A i + 1 Δt Δt l=1,N l i δ l,i M l,i c,s,out ( t )dt = l=1,N l i ( 1 δ l,i ) M ¯ l,i c,s,in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadghajuaGdaWgaaqcbasaaKqzadGaam4Caiaa cYcacaWGPbaaleqaaKqbaoaabmaakeaajugibiaadshaaOGaayjkai aawMcaaaqaaKqzGeGaamizaiaadshaaaGaamyqaKqbaoaaBaaajeai baqcLbmacaWGPbaaleqaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGe GaaGymaaGcbaqcLbsacqqHuoarcaWG0baaaKqbaoaapefakeaajuaG daaeqbGcbaqcLbsacqaH0oazjuaGdaWgaaqcbasaaKqzadGaamiBai aacYcacaWGPbaaleqaaKqzGeGaamytaSWaa0baaKqaGeaajugWaiaa dYgacaGGSaGaamyAaaqcbasaaKqzadGaam4yaiaacYcacaWGZbGaai ilaiaad+gacaWG1bGaamiDaaaajuaGdaqadaGcbaqcLbsacaWG0baa kiaawIcacaGLPaaajugibiaadsgacaWG0baajeaibaqcLbmacaWGSb Gaeyypa0JaaGymaiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugW aiaadMgaaKGaGeqaaaWcbeqcLbsacqGHris5aaWcbaqcLbsacqqHuo arcaWG0baaleqajugibiabgUIiYdGaeyypa0tcfa4aaabuaOqaaKqb aoaabmaakeaajugibiaaigdacqGHsislcqaH0oazjuaGdaWgaaqcba saaKqzadGaamiBaiaacYcacaWGPbaaleqaaaGccaGLOaGaayzkaaqc LbsaceWGnbGbaebalmaaDaaajeaibaqcLbmacaWGSbGaaiilaiaadM gaaKqaGeaajugWaiaadogacaGGSaGaam4CaiaacYcacaWGPbGaamOB aaaaaKqaGeaajugWaiaadYgacqGH9aqpcaaIXaGaaiilaiaad6eaca WGSbWcdaWgaaqccasaaKqzadGaamyAaaqccasabaaaleqajugibiab ggHiLdaaaa@9EA7@  s = x, y (27),

with F l,i c,out =max( 0, F l,i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb WcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbma caWGJbGaaiilaiaad+gacaWG1bGaamiDaaaajugibiabg2da9iGac2 gacaGGHbGaaiiEaKqbaoaabmaakeaajugibiaaicdacaGGSaGaamOr aKqbaoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqabaaaki aawIcacaGLPaaaaaa@4F32@   M l,i c,s,out = F l,i c,out q s,i / h i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb WcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbma caWGJbGaaiilaiaadohacaGGSaGaam4BaiaadwhacaWG0baaaKqzGe Gaeyypa0JaamOraSWaa0baaKqaGeaajugWaiaadYgacaGGSaGaamyA aaqcbasaaKqzadGaam4yaiaacYcacaWGVbGaamyDaiaadshaaaqcfa 4aaSGbaOqaaKqzGeGaamyCaKqbaoaaBaaajeaibaqcLbmacaWGZbGa aiilaiaadMgaaSqabaaakeaajugibiaadIgajuaGdaWgaaqcbasaaK qzadGaamyAaaWcbeaaaaaaaa@5A94@   s = x, y if ϕ i k < ϕ ip k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaKqz GeGaeyipaWJaeqy1dy2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaK qaGeaajugWaiaadUgaaaaaaa@4611@        (28,a),

and F l,i c,in =min( 0, F l,i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aa0baaKqaGeaajugWaiaadYgacaGGSaGaamyAaaqcbasaaKqz adGaam4yaiaacYcacaWGPbGaamOBaaaajugibiabg2da9iGac2gaca GGPbGaaiOBaKqbaoaabmaakeaajugibiaaicdacaGGSaGaamOraKqb aoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqabaaakiaawI cacaGLPaaaaaa@4EAD@   M l,i c,s,in = F l,i c,in q s,ip / h ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb WcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbma caWGJbGaaiilaiaadohacaGGSaGaamyAaiaad6gaaaqcLbsacqGH9a qpcaWGgbWcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeai baqcLbmacaWGJbGaaiilaiaadMgacaWGUbaaaKqbaoaalyaakeaaju gibiaadghajuaGdaWgaaqcbasaaKqzadGaam4CaiaacYcacaWGPbGa amiCaaWcbeaaaOqaaKqzGeGaamiAaKqbaoaaBaaajeaibaqcLbmaca WGPbGaamiCaaWcbeaaaaaaaa@5A72@  if ϕ i k ϕ ip k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaKqz GeGaeyyzImRaeqy1dy2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaK qaGeaajugWaiaadUgaaaaaaa@46D3@  (28,b).
The source terms is allocated in the convective prediction step [32]. The initial state in each cell of the convective correction step is the one computed in the same cell at the end of the convective prediction step. After the ODEs system (26)-(27) is solved in cell i, the mean-in-time value of the flux/momentum flux leaving from side l of i to the neighboring cell ip with ϕ i k < ϕ ip k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaKqz GeGaeyipaWJaeqy1dy2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaK qaGeaajugWaiaadUgaaaaaaa@4611@ , from tk+1/3 to tk+2/3, is computed similarly to the previous CP step. Further comments concerning the MAST procedure can be found in [31,32].

The mean-in-time values h ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmiAay aaraaaaa@3699@  and ( Uh ) 2 + ( Vh ) 2 h 7/3 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqbaoaanaaake aajuaGdaGcaaGcbaqcfa4aaeWaaOqaaKqzGeGaamyvaiaadIgaaOGa ayjkaiaawMcaaKqbaoaaCaaaleqajqwaa+FaaKqzadGaaGOmaaaaju gibiabgUcaRKqbaoaabmaakeaajugibiaadAfacaWGObaakiaawIca caGLPaaajuaGdaahaaWcbeqcKfaG=haajugWaiaaikdaaaGcdGafaU GaaeacuaiabGafacaajugibiacGa4GObqcfa4aiacGCaaaleqcGayc basaiacGjugWaiacGaiI3aGaiacGc+cacGaiaI4maaaaaSqabaaaaa aa@5640@ , required for the DC step (see Eqs. (16)), are obtained by numerical integration during the solution of each ODEs system during the convective prediction and correction steps [31,32].

The diffusive correction problem

After the prediction components of the water depths (levels) are computed with piecewise constant approximation (P0) inside each cell, a continuous piecewise linear (P1) shape is assigned to the water depths (levels) computed at the end of the (CP + CC) steps, obtained from the nodal values. If we move from the P0 to the P1 approximation, at a given time level, we do not alter the estimation of the global mass, if the polygons of the P0 approximation (Figure 2b) are the same as the ones of the P1 approximation. If we use, for the solution of the DC problem the conforming P1 Galerkin FE scheme, where the polygons are delimited by the centers of mass (see Appendix C), we introduce a small mass balance error. We avoid this error, using the same polygons for the P0 and P1 approximations. Other details in Appendix C.

We use a fully implicit time discretization of the DC problem (15)-(16) [32]. Under the hypothesis of fixed bed condition, we get the following differential equation for the generic cell i

H ˜ i k+1 H i k+2/3 Δt + ( U ˜ h ˜ ) k+1 x + ( V ˜ h ˜ ) k+1 y = ( Uh ) ¯ x + ( Vh ) ¯ y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGabmisayaaiaWcdaqhaaqcbasaaKqzadGaamyAaaqcbasa aKqzadGaam4AaiabgUcaRiaaigdaaaqcLbsacqGHsislcaWGibWcda qhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaam4AaiabgUcaRiaa ikdacaGGVaGaaG4maaaaaOqaaKqzGeGaeuiLdqKaamiDaaaacqGHRa WkjuaGdaWcaaGcbaqcLbsacqGHciITjuaGdaqadaGcbaqcLbsaceWG vbGbaGaaceWGObGbaGaaaOGaayjkaiaawMcaaKqbaoaaCaaaleqaje aibaqcLbmacaWGRbGaey4kaSIaaGymaaaaaOqaaKqzGeGaeyOaIyRa amiEaaaacqGHRaWkjuaGdaWcaaGcbaqcLbsacqGHciITjuaGdaqada GcbaqcLbsaceWGwbGbaGaaceWGObGbaGaaaOGaayjkaiaawMcaaKqb aoaaCaaaleqajeaibaqcLbmacaWGRbGaey4kaSIaaGymaaaaaOqaaK qzGeGaeyOaIyRaamyEaaaacqGH9aqpjuaGdaWcaaGcbaqcLbsacqGH ciITjuaGdaqdaaGcbaqcfa4aaeWaaOqaaKqzGeGaamyvaiaadIgaaO GaayjkaiaawMcaaaaaaeaajugibiabgkGi2kaadIhaaaGaey4kaSsc fa4aaSaaaOqaaKqzGeGaeyOaIyBcfa4aa0aaaOqaaKqbaoaabmaake aajugibiaadAfacaWGObaakiaawIcacaGLPaaaaaaabaqcLbsacqGH ciITcaWG5baaaaaa@823B@  (29),

q ˜ s,i k+1 q ˜ s,i k+2/3 Δt +g h ¯ i H ˜ k+1 s +g q ˜ s,i k+1 so r i =g h ¯ i H k s +g q ¯ s,i so r i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabiqaciaacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaa GcbaqcLbsaceWGXbGbaGaalmaaDaaajeaibaqcLbmacaWGZbGaaiil aiaadMgaaKqaGeaajugWaiaadUgacqGHRaWkcaaIXaaaaKqzGeGaaG jcVlabgkHiTiaayIW7ceWGXbGbaGaalmaaDaaajeaibaqcLbmacaWG ZbGaaiilaiaadMgaaKqaGeaajugWaiaadUgacqGHRaWklmaalyaaje aibaqcLbmacaaIYaaajeaibaqcLbmacaaIZaaaaaaaaOqaaKqzGeGa euiLdqKaamiDaaaacqGHRaWkcaWGNbGaaGPaVlqadIgagaqeaKqbao aaBaaajeaibaqcLbmacaWGPbaaleqaaKqbaoaalaaakeaajugibiab gkGi2kqadIeagaacaKqbaoaaCaaaleqajeaibaqcLbmacaWGRbGaey 4kaSIaaGymaaaaaOqaaKqzGeGaeyOaIyRaam4CaaaacaaMi8Uaey4k aSIaaGPaVlaayIW7caWGNbGaaGPaVlqadghagaacaSWaa0baaKqaGe aajugWaiaadohacaGGSaGaamyAaaqcbasaaKqzadGaam4AaiabgUca RiaaigdaaaqcLbsacaWGZbGaam4BaiaadkhajuaGdaWgaaqcbasaaK qzadGaamyAaaWcbeaajugibiabg2da9iaadEgacaaMc8UabmiAayaa raqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaqcfa4aaSaaaOqaaK qzGeGaeyOaIyRaamisaKqbaoaaCaaaleqajeaibaqcLbmacaWGRbaa aaGcbaqcLbsacqGHciITcaWGZbaaaiaayIW7cqGHRaWkcaWGNbGaaG PaVlqadghagaqeaKqbaoaaBaaajeaibaqcLbmacaWGZbGaaiilaiaa dMgaaSqabaqcLbsacaaMc8Uaam4Caiaad+gacaWGYbqcfa4aaSbaaK qaGeaajugWaiaadMgaaSqabaaaaa@A2BC@  s = x, y (30),
where so r i = n i 2 ( ( Uh ) i 2 + ( Vh ) i 2 / h i 7/3 ) ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabiqaciaacaGaaeqabaWaaeaaeaaakeaajugibiaado hacaWGVbGaamOCaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqz GeGaaGjcVlabg2da9iaayIW7caWGUbWcdaqhaaqcbasaaKqzadGaam yAaaqcbasaaKqzadGaaGOmaaaajuaGdaqdaaGcbaqcfa4aaeWaaOqa aKqbaoaalyaakeaajuaGdaGcaaGcbaqcfa4aaeWaaOqaaKqzGeGaam yvaiaadIgaaOGaayjkaiaawMcaaSWaa0baaKqaGeaajugWaiaadMga aKqaGeaajugWaiaaikdaaaqcLbsacaaMi8Uaey4kaSIaaGjcVNqbao aabmaakeaajugibiaadAfacaWGObaakiaawIcacaGLPaaalmaaDaaa jeaibaqcLbmacaWGPbaajeaibaqcLbmacaaIYaaaaaWcbeaaaOqaaK qzGeGaamiAaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWaiaa iEdacaGGVaGaaG4maaaaaaaakiaawIcacaGLPaaaaaaaaa@6957@ . From Eq. (30), the flow rate components at the end of the hydrostatic step are

q ˜ s,i k+1 = D ˜ i H ˜ k+1 s + k i s + q s,i k+2/3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGXb GbaGaalmaaDaaajeaibaqcLbmacaWGZbGaaiilaiaadMgaaKqaGeaa jugWaiaadUgacqGHRaWkcaaIXaaaaKqzGeGaeyypa0JaeyOeI0Iabm irayaaiaqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaqcfa4aaSaa aOqaaKqzGeGaeyOaIyRabmisayaaiaqcfa4aaWbaaSqabKqaGeaaju gWaiaadUgacqGHRaWkcaaIXaaaaaGcbaqcLbsacqGHciITcaWGZbaa aiabgUcaRiaadUgalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLb macaWGZbaaaKqzGeGaey4kaSIaamyCaSWaa0baaKqaGeaajugWaiaa dohacaGGSaGaamyAaaqcbasaaKqzadGaam4AaiabgUcaRiaaikdaca GGVaGaaG4maaaaaaa@635F@  s = x, y (31),

with coefficients D ˜ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGeb GbaGaajuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaaaaa@3A5B@  and k i x( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb WcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaamiEaSWaaeWa aKqaGeaajugWaiaadMhaaKqaGiaawIcacaGLPaaaaaaaaa@404F@  defined as

D ˜ i ={ g h ¯ i Δt/ [ 1+Δtgso r i ]      if    ϕ i k ϕ ip k g h ¯ ip Δt/ [ 1+Δtgso r ip ]  if    ϕ ip k > ϕ i k             k i s = D ˜ i H k s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGeb GbaGaajuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajugibiabg2da 9KqbaoaaceaajugibqaabeGcbaqcfa4aaSGbaOqaaKqzGeGaam4zai qadIgagaqeaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqzGeGa euiLdqKaamiDaaGcbaqcfa4aamWaaOqaaKqzGeGaaGymaiabgUcaRi abfs5aejaadshacaaMc8Uaam4zaiaaykW7caaMc8Uaam4Caiaad+ga caWGYbqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaaakiaawUfaca GLDbaaaaqcLbsacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabMga caqGMbGaaeiiaiaabccacaqGGaGaeqy1dy2cdaqhaaqcbasaaKqzad GaamyAaaqcbasaaKqzadGaam4AaaaajugibiabgwMiZkabew9aMTWa a0baaKqaGeaajugWaiaadMgacaWGWbaajeaibaqcLbmacaWGRbaaaa Gcbaqcfa4aaSGbaOqaaKqzGeGaam4zaiqadIgagaqeaKqbaoaaBaaa jeaibaqcLbmacaWGPbGaamiCaaWcbeaajugibiabfs5aejaadshaaO qaaKqbaoaadmaakeaajugibiaaigdacqGHRaWkcqqHuoarcaWG0bGa aGPaVlaadEgacaaMc8UaaGPaVlaadohacaWGVbGaamOCaKqbaoaaBa aajeaibaqcLbmacaWGPbGaamiCaaWcbeaaaOGaay5waiaaw2faaaaa jugibiaabccacaqGPbGaaeOzaiaabccacaqGGaGaaeiiaiabew9aML qbaoaaDaaajeaibaqcLbmacaWGPbGaamiCaaqcbasaaKqzadGaam4A aaaajugibiabg6da+iabew9aMLqbaoaaDaaajeaibaqcLbmacaWGPb aajeaibaqcLbmacaWGRbaaaaaakiaawUhaaKqzGeGaaGPaVlaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaam4AaKqbaoaaDaaajeaibaqcLbmacaWGPbaajeai baqcLbmacaWGZbaaaKqzGeGaeyypa0Jabmirayaaiaqcfa4aaSbaaK qaGeaajugWaiaadMgaaSqabaqcfa4aaSaaaOqaaKqzGeGaeyOaIyRa amisaKqbaoaaCaaaleqajeaibaqcLbmacaWGRbaaaaGcbaqcLbsacq GHciITcaWGZbaaaaaa@C316@  s = x, y (32).

We merge Eqs. (31) in Eq. (29) and, after space integration and application of the Green's theorem, we get the following balance law for cell i

A i H i t d A i l=1,N l i ( D ˜ i H ˜ k+1 n i,ip d l,i ) = l=1,N l i ( D ˜ i H ˜ k n i,ip d l,i ) + l=1,N l i ( q ¯ i q i k+2/3 ) n ^ i,ip d l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqVepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaajuaGdaWdrb Gcbaqcfa4aaSaaaOqaaKqzGeGaeyOaIyRaamisaKqbaoaaBaaajeai baqcLbmacaWGPbaaleqaaaGcbaqcLbsacqGHciITcaWG0baaaiaads gacaWGbbqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaaajeaibaqc LbmacaWGbbWcdaWgaaqccasaaKqzadGaamyAaaqccasabaaaleqaju gibiabgUIiYdGaeyOeI0scfa4aaabuaOqaaKqbaoaabmaakeaajugi biqadseagaacaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqbao aalaaakeaajugibiabgkGi2kqadIeagaacaKqbaoaaCaaaleqajeai baqcLbmacaWGRbGaey4kaSIaaGymaaaaaOqaaKqzGeGaeyOaIyRaam OBaKqbaoaaBaaajeaibaqcLbmacaWGPbGaaiilaiaadMgacaWGWbaa leqaaaaajugibiaadsgajuaGdaWgaaqcbasaaKqzadGaamiBaiaacY cacaWGPbaaleqaaaGccaGLOaGaayzkaaaajeaibaqcLbmacaWGSbGa eyypa0JaaGymaiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugWai aadMgaaKGaGeqaaaWcbeqcLbsacqGHris5aiabg2da9iabgkHiTKqb aoaaqafakeaajuaGdaqadaGcbaqcLbsaceWGebGbaGaajuaGdaWgaa qcbasaaKqzadGaamyAaaWcbeaajuaGdaWcaaGcbaqcLbsacqGHciIT ceWGibGbaGaajuaGdaahaaWcbeqcbasaaKqzadGaam4AaaaaaOqaaK qzGeGaeyOaIyRaamOBaKqbaoaaBaaajeaibaqcLbmacaWGPbGaaiil aiaadMgacaWGWbaaleqaaaaajugibiaadsgajuaGdaWgaaqcbasaaK qzadGaamiBaiaacYcacaWGPbaaleqaaaGccaGLOaGaayzkaaaajeai baqcLbmacaWGSbGaeyypa0JaaGymaiaacYcacaWGobGaamiBaSWaaS baaKGaGeaajugWaiaadMgaaKGaGeqaaaWcbeqcLbsacqGHris5aiab gUcaRKqbaoaaqafakeaajuaGdaqadaGcbaqcLbsaceWHXbGbaebaju aGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajugibiabgkHiTiaahgha lmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbGaey4kaS IaaGOmaiaac+cacaaIZaaaaaGccaGLOaGaayzkaaqcLbsacqGHflY1 ceWHUbGbaKaajuaGdaWgaaqcbasaaKqzadGaamyAaiaacYcacaWGPb GaamiCaaWcbeaajugibiaadsgajuaGdaWgaaqcbasaaKqzadGaamiB aiaacYcacaWGPbaaleqaaaqcbasaaKqzadGaamiBaiabg2da9iaaig dacaGGSaGaamOtaiaadYgalmaaBaaajiaibaqcLbmacaWGPbaajiai beaaaSqabKqzGeGaeyyeIuoaaaa@CEE3@  (33),
where H/ n i,ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaamaalyaabaqcLb sacqGHciITcaWGibaakeaajugibiabgkGi2kaad6gajuaGdaWgaaqc basaaKqzadGaamyAaiaacYcacaWGPbGaamiCaaWcbeaaaaaaaa@4061@  is the water level gradient along the n ^ i,ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWHUb GbaKaajuaGdaWgaaqcbasaaKqzadGaamyAaiaacYcacaWGPbGaamiC aaWcbeaaaaa@3D1D@  direction and the other symbols are specified as above. Concerning the second term on the r.h.s. of Eq. (33), we have

l=1,N l i ( q ¯ i n ^ i,ip d l,i ) = F ¯ i in F ¯ i out = H i k+2/3 H i k Δt A i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqbaoaaqafake aajuaGdaqadaGcbaqcLbsacaaMc8UabCyCayaaraWcdaWgaaqcbasa aKqzadGaamyAaaqcbasabaqcLbsacqGHflY1ceWHUbGbaKaajuaGda WgaaqcbasaaKqzadGaamyAaiaacYcacaWGPbGaamiCaaWcbeaajugi biaadsgajuaGdaWgaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaale qaaaGccaGLOaGaayzkaaaajeaibaqcLbmacaWGSbGaeyypa0JaaGym aiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugWaiaadMgaaKGaGe qaaaWcbeqcLbsacqGHris5aiabg2da9iqadAeagaqeaSWaa0baaKqa GeaajugWaiaadMgaaKqaGeaajugWaiaadMgacaWGUbaaaKqzGeGaey OeI0IabmOrayaaraWcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqz adGaam4BaiaadwhacaWG0baaaOGaeyypa0ZaaSaaaeaajugibiaadI ealmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbGaey4k aSIaaGOmaiaac+cacaaIZaaaaKqzGeGaeyOeI0IaamisaSWaa0baaK qaGeaajugWaiaadMgaaKqaGeaajugWaiaadUgaaaaakeaacqqHuoar caWG0baaaiaadgeadaWgaaWcbaGaamyAaaqabaaaaa@7F14@  (34),
while q i k+2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqqFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWHXb WcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaam4AaiabgUca RiaaikdacaGGVaGaaG4maaaaaaa@3F07@  is computed after the (CP + CC) steps. After discretization in time, Eq. (33) becomes

H ˜ i k+1 H i k+2/3 Δt A i + l=1,N l i F ˜ l,i = l=1,N l i b ˜ l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaamaalaaajaaiba GabmisayaaiaGcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGa am4AaiabgUcaRiaaigdaaaqcaaIaeyOeI0IaamisaSWaa0baaKqaGe aajugWaiaadMgaaKqaGeaajugWaiaadUgacqGHRaWkcaaIYaGaai4l aiaaiodaaaaajaaibaGaeuiLdqKaamiDaaaajugibiaadgealmaaBa aajeaibaqcLbmacaWGPbaajeaibeaajugibiabgUcaRKqbaoaaqafa keaajugibiqadAeagaacaKqbaoaaBaaajeaibaqcLbmacaWGSbGaai ilaiaadMgaaSqabaaajeaibaqcLbmacaWGSbGaeyypa0JaaGymaiaa cYcacaWGobGaamiBaSWaaSbaaKGaGeaajugWaiaadMgaaKGaGeqaaa WcbeqcLbsacqGHris5aiabg2da9Kqbaoaaqafakeaajugibiqadkga gaacaKqbaoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqaba aajeaibaqcLbmacaWGSbGaeyypa0JaaGymaiaacYcacaWGobGaamiB aSWaaSbaaKGaGeaajugWaiaadMgaaKGaGeqaaaWcbeqcLbsacqGHri s5aaaa@73EE@  i = 1, ..., N (35),

where F ˜ l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgea0lrP0xe9Fve9 Fve9VXqaaeGabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3E66@  is the corrective flux across side l of cell i

F ˜ l,i = D ˜ i H ˜ k+1 n i,ip d l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa jugibiabg2da9iabgkHiTiqadseagaacaKqbaoaaBaaajeaibaqcLb macaWGPbaaleqaaOWaaSaaaeaajugibiabgkGi2kqadIeagaacaKqb aoaaCaaaleqajeaibaqcLbmacaWGRbGaey4kaSIaaGymaaaaaOqaaK qzGeGaeyOaIyRaamOBaKqbaoaaBaaajeaibaqcLbmacaWGPbGaaiil aiaadMgacaWGWbaaleqaaaaajugibiaadsgajuaGdaWgaaqcKfaG=h aajugWaiaadYgacaGGSaGaamyAaaqcbawabaaaaa@59B3@                                                                            (36.a),

and the term b ˜ l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgea0lrP0xe9Fve9 Fve9VXqaaeGabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOyay aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3E82@  on the r.h.s. of Eq. (35) is

b ˜ l,i = D ˜ i H ˜ k n i,ip d l,i +( q ¯ i q i k+2/3 ) n ^ i,ip i,ip d l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOyay aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa jugibiabg2da9iabgkHiTiqadseagaacaKqbaoaaBaaajeaibaqcLb macaWGPbaaleqaaOWaaSaaaeaajugibiabgkGi2kqadIeagaacaKqb aoaaCaaaleqajeaibaqcLbmacaWGRbaaaaGcbaqcLbsacqGHciITca WGUbqcfa4aaSbaaKqaGeaajugWaiaadMgacaGGSaGaamyAaiaadcha aSqabaaaaKqzGeGaamizaKqbaoaaBaaajeaibaqcLbmacaWGSbGaai ilaiaadMgaaSqabaqcLbsacqGHRaWkjuaGdaqadaGcbaqcLbsaceWH XbGbaebajuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajugibiabgk HiTiaahghalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWG RbGaey4kaSIaaGOmaiaac+cacaaIZaaaaaGccaGLOaGaayzkaaqcLb sacqGHflY1ceWHUbGbaKaajuaGdaWgaaqcbasaaKqzadGaamyAaiaa cYcacaWGPbGaamiCaaWcbeaajuaGdaWgaaWcbaqcLbsacaWGPbGaai ilaiaadMgacaWGWbaaleqaaKqzGeGaamizaKqbaoaaBaaajeaibaqc LbmacaWGSbGaaiilaiaadMgaaSqabaaaaa@7CED@  (36.b).
Due to the P1 spatial approximation, H ˜ k+1 / n i,ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaamaalyaabaqcLb sacqGHciITceWGibGbaGaajuaGdaahaaWcbeqcbasaaKqzadGaam4A aiabgUcaRiaaigdaaaaakeaajugibiabgkGi2kaad6gajuaGdaWgaa qcbasaaKqzadGaamyAaiaacYcacaWGPbGaamiCaaWcbeaaaaaaaa@4510@  only depends on the values at nodes i and ip (since n ^ i,ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiqah6 gagaqcaKqbaoaaBaaajeaibaqcLbmacaWGPbGaaiilaiaadMgacaWG WbaaleqaaKqzGeGaaGjcVdaa@3F55@ is parallel to ri,ip and orthogonal to side l, see sections 3.2 and 4.1), and F ˜ l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgea0lrP0xe9Fve9 Fve9VXqaaeGabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3E66@  is written

F ˜ l,i = D ˜ i ( H ˜ ip k+1 H ˜ i k+1 ) | r i,ip | d l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa jugibiabg2da9iabgkHiTiqadseagaacaKqbaoaaBaaajeaibaqcLb macaWGPbaaleqaaKqbaoaabmaakeaajugibiqadIeagaacaSWaa0ba aKqaGeaajugWaiaadMgacaWGWbaajeaibaqcLbmacaWGRbGaey4kaS IaaGymaaaajugibiabgkHiTiqadIeagaacaSWaa0baaKqaGeaajugW aiaadMgaaKqaGeaajugWaiaadUgacqGHRaWkcaaIXaaaaaGccaGLOa GaayzkaaWaaSGaaeaaaeaaaaqcLbsacaaMe8Ecfa4aaqWaaOqaaKqz GeGaaCOCaKqbaoaaBaaajeaibaqcLbmacaWGPbGaaiilaiaadMgaca WGWbaaleqaaaGccaGLhWUaayjcSdqcLbsacaWGKbqcfa4aaSbaaKqa GeaajugWaiaadYgacaGGSaGaamyAaaWcbeaaaaa@6823@  (37).

Eqs. (35) constitute a linear system of order N in the Hi (i = 1, …, N) unknowns, whose matrix is sparse and symmetric. Generally the non-zero entries are about 5% or less of the total number of matrix coefficients. The diagonal and off-diagonal matrix coefficients are, respectively

s i,i = A i Δt + l=1,N l i D ˜ i | r i,ip | d l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGaam4CaK qbaoaaBaaajeaibaqcLbmacaWGPbGaaiilaiaadMgaaSqabaGccqGH 9aqpdaWcaaqaaKqzGeGaamyqaKqbaoaaBaaajeaibaqcLbmacaWGPb aaleqaaaGcbaqcLbsacqqHuoarcaWG0baaaOGaey4kaSYaaabuaeaa daWcaaqaaKqzGeGabmirayaaiaqcfa4aaSbaaKqaGeaajugWaiaadM gaaSqabaaakeaajuaGdaabdaGcbaqcLbsacaWHYbqcfa4aaSbaaKqa GeaajugWaiaadMgacaGGSaGaamyAaiaadchaaSqabaaakiaawEa7ca GLiWoaaaqcLbsacaWGKbqcfa4aaSbaaKqaGeaajugWaiaadYgacaGG SaGaamyAaaWcbeaaaKqaGeaajugWaiaadYgacqGH9aqpcaaIXaGaai ilaiaad6eacaWGSbWcdaWgaaqccasaaKqzadGaamyAaaqccasabaaa leqaniabggHiLdGccaaMc8oaaa@66CF@   s i,ip = D ˜ i | r i,ip | d l,i δ i,ip MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGaam4CaK qbaoaaBaaajeaibaqcLbsacaWGPbGaaiilaiaadMgacaWGWbaaleqa aKqzGeGaeyypa0JaeyOeI0IcdaWcaaqcfayaaiqadseagaacamaaBa aajuaibaGaamyAaaqcfayabaaabaWaaqWaaeaacaWHYbWaaSbaaKqb GeaacaWGPbGaaiilaiaadMgacaWGWbaajuaGbeaaaiaawEa7caGLiW oaaaqcLbwacaaMc8EcLbsacaaMi8UaamizaKqbaoaaBaaajeaibaqc LbmacaWGSbGaaiilaiaadMgaaSqabaqcLbsacaaMc8UaeqiTdqwcfa 4aaSbaaKqaGeaajugWaiaadMgacaGGSaGaamyAaiaadchaaSqabaaa aa@5D1C@  i, ip = 1, ..., N (38),

where δ i ,ip = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH0oazl8aadaWgaaqcbasaaKqzadWdbiaadMgaaKqaG8aa beaalmaaBaaajeaibaqcLbmapeGaaiilaiaadMgacaWGWbaajeaipa qabaGcpeGaeyypa0Jaaeiiaiaaigdaaaa@4242@  if nodes i and ip are linked, otherwise it is equal to zero.
According to Eqs. (37)-(38), the off-diagonal coefficients are non-negative if the GD mesh property is not satisfied for side ri,ip and the M-matrix property is lost. In this case, dl,i is negative (see Eq. (10) and Appendix B) and the sign of F ˜ l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3B0F@  is not consistent with the difference ( H ˜ ip k+1 H ˜ i k+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiqadIeagaacaSWaa0baaKqaGeaajugWaiaadMgacaWGWbaa jeaibaqcLbmacaWGRbGaey4kaSIaaGymaaaajugibiabgkHiTiqadI eagaacaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWaiaadUga cqGHRaWkcaaIXaaaaaGccaGLOaGaayzkaaaaaa@4898@  in Eq. (37). On the opposite, if side ri,ip satisfies the GD property, dl,i is positive, and the sign of the flux in Eq. (37) only depends on the difference of the water levels in cells i and ip. If the GD property is satisfied for all the sides of the mesh, the matrix of system (35) is a M-matrix, positive-definite and, according to [41], the system is well-conditioned.

The numerical scheme adopted for the solution of the DC problem has a structure similar to the P1 conforming FE Galerkin scheme, and we present and discuss analogies and differences between the two numerical formulations in Appendix C.

We solve system (35) by a preconditioned conjugate gradient method (with incomplete Cholesky preconditioning). Due to the symmetry of the matrix of system (35), we only save the non-zero entries of the lower triangular matrix, using a Symmetric Coordinate Storage algorithm [43 ].

After the solution of system (35) for the H unknowns, we compute the gradients of the water level x( y ) H ˜ i k+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrpepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiabgE GirNqbaoaaBaaaleaajugibiaadIhajuaGdaqadaWcbaqcLbsacaWG 5baaliaawIcacaGLPaaaaeqaaKqzGeGabmisayaaiaWcdaqhaaqcba saaKqzadGaamyAaaqcbasaaKqzadGaam4AaiabgUcaRiaaigdaaaaa aa@467F@  at the end of the DC step as shown in the next section. As specified in section 4.1, the values of the gradients of the water level are assumed constant during the solution of the two convective steps of the hydrostatic problem in the next time iteration [31,32]. Finally, the components of the flow rate q ˜ s,i k+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGXb GbaGaalmaaDaaajeaibaqcLbmacaWGZbGaaiilaiaadMgaaKqaGeaa jugWaiaadUgacqGHRaWkcaaIXaaaaaaa@3F89@  (s = x, y) are updated according to Eqs. (31)-(32).

According to Eq. (37) and the definition of the coefficient D ˜ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGeb GbaGaajuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaaaaa@3A5B@  in (Eq. (32)), the flux F ˜ l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgea0lrP0xe9Fve9 Fve9VXqaaeGabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3E66@  leaving from cell i (ip) to ip (i) is equal to the flux entering cell ip (i) from cell i (ip). If F ˜ l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgea0lrP0xe9Fve9 Fve9VXqaaeGabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3E66@  is leaving from i to ip or vice versa depends on the difference ( H ˜ i k+1 H ˜ ip k+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiqadIeagaacaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaa jugWaiaadUgacqGHRaWkcaaIXaaaaKqzGeGaeyOeI0Iabmisayaaia WcdaqhaaqcbasaaKqzadGaamyAaiaadchaaKqaGeaajugWaiaadUga cqGHRaWkcaaIXaaaaaGccaGLOaGaayzkaaaaaa@4898@  (see Eq. (37).

Computation of the spatial gradients of the water level

We compute the gradient of H in cell i as follows. Call n ^ q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWHUb GbaKaajuaGdaWgaaqcbasaaKqzadGaamyCaaWcbeaaaaa@3A3E@  the unit vector parallel to the flow vector in cell i. Let H i,ip MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqGHhi s0caWGibqcfa4aaSbaaKqaGeaajugWaiaadMgacaGGSaGaamyAaiaa dchaaSqabaaaaa@3E15@  be the vector computed as (ip is one of the cells linked with i),

H i,ip = ( H i H ip )/ r i,ip MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqGHhi s0caWGibqcfa4aaSbaaKqaGeaajugWaiaadMgacaGGSaGaamyAaiaa dchaaSqabaqcLbsacqGH9aqpjuaGdaWcgaGcbaqcfa4aaeWaaOqaaK qzGeGaamisaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqzGeGa eyOeI0IaamisaKqbaoaaBaaajeaibaqcLbmacaWGPbGaamiCaaWcbe aaaOGaayjkaiaawMcaaaqaaKqzGeGaaCOCaKqbaoaaBaaajeaibaqc LbmacaWGPbGaaiilaiaadMgacaWGWbaaleqaaaaaaaa@543A@ (39).

We assume

H i = ip=1,N l i max( 0,( n ^ i,ip n ^ q ) H i,ip ) ip=1,N l i max( 0,( n ^ i,ip n ^ q ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqGHhi s0caWGibqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcfa4aaabeaOqaaKqzGeGaciyBaiaacggaca GG4bqcfa4aaeWaaOqaaKqzGeGaaGimaiaacYcajuaGdaqadaGcbaqc LbsaceWHUbGbaKaajuaGdaWgaaqcbasaaKqzadGaamyAaiaacYcaca WGPbGaamiCaaWcbeaajugibiabgwSixlqah6gagaqcaKqbaoaaBaaa jeaibaqcLbmacaWGXbaaleqaaaGccaGLOaGaayzkaaqcLbsacqGHhi s0caWGibqcfa4aaSbaaKqaGeaajugWaiaadMgacaGGSaGaamyAaiaa dchaaSqabaaakiaawIcacaGLPaaaaKqaGeaajugWaiaadMgacaWGWb Gaeyypa0JaaGymaiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugW aiaadMgaaKGaGeqaaaWcbeqcLbsacqGHris5aaGcbaqcfa4aaabeaO qaaKqzGeGaciyBaiaacggacaGG4bqcfa4aaeWaaOqaaKqzGeGaaGim aiaacYcajuaGdaqadaGcbaqcLbsaceWHUbGbaKaajuaGdaWgaaqcba saaKqzadGaamyAaiaacYcacaWGPbGaamiCaaWcbeaajugibiabgwSi xlqah6gagaqcaKqbaoaaBaaajeaqbaqcLboacaWGXbaaleqaaaGcca GLOaGaayzkaaaacaGLOaGaayzkaaaajeaibaqcLbmacaWGPbGaamiC aiabg2da9iaaigdacaGGSaGaamOtaiaadYgalmaaBaaajiaibaqcLb macaWGPbaajiaibeaaaSqabKqzGeGaeyyeIuoaaaaaaa@9010@ (40),

where n ^ i,ip n ^ q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWHUb GbaKaajuaGdaWgaaqcbasaaKqzadGaamyAaiaacYcacaWGPbGaamiC aaWcbeaajugibiabgwSixlqah6gagaqcaKqbaoaaBaaajeaibaqcLb macaWGXbaaleqaaaaa@43B1@  is the dot product between the direction of side ri,ip and the unitary flow rate vector. If cell i lies on an impervious boundary, the flow rate vector is parallel to the boundary and H i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqGHhi s0caWGibqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaaaaa@3B82@ , computed as in Eq. (40), is projected along the boundary direction.

The expression in Eq. (40) can be regarded as an upwind formulation, where H i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqGHhi s0caWGibqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaaaaa@3B82@  depends on the differences between Hi and the values of H in the downstream (along the flow direction) ip cells. This is consistent with the procedure adopted in the (CP + CC) steps, where the unknown fluxes/momentum fluxes are the ones leaving from i to the downstream ip cells.

The Hydrostatic Problem

Eqs. (8)-(9) are integrated in space (see also Eqs. (A.3)-(A.5) in Appendix A), and, using the Green’s theorem, we get the integral formulation

h i t A i + l=1,N l i F l,i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabiqaciaacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaa GcbaqcLbsacqGHciITcaWGObqcfa4aaSbaaKqaGeaajugWaiaadMga aSqabaaakeaajugibiabgkGi2kaadshaaaGaamyqaKqbaoaaBaaaje aibaqcLbmacaWGPbaaleqaaKqzGeGaey4kaSscfa4aaabuaOqaaKqz GeGaamOraKqbaoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaS qabaaajeaibaqcLbmacaWGSbGaeyypa0JaaGymaiaacYcacaWGobGa amiBaSWaaSbaaKGaGeaajugWaiaadMgaaKGaGeqaaaWcbeqcLbsacq GHris5aiabg2da9iaaicdaaaa@5926@  i = 1, …, N                                               (13),

q s,i t A i + l=1,N l i M l,i s + R i s =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabiqaciaacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaa GcbaqcLbsacqGHciITcaWGXbqcfa4aaSbaaKqaGeaajugWaiaadoha caGGSaGaamyAaaWcbeaaaOqaaKqzGeGaeyOaIyRaamiDaaaacaWGbb qcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaqcLbsacqGHRaWkjuaG daaeqbGcbaqcLbsacaaMc8UaamytaSWaa0baaKqaGeaajugWaiaadY gacaGGSaGaamyAaaqcbasaaKqzadGaam4CaaaaaKqaGeaajugWaiaa dYgacqGH9aqpcaaIXaGaaiilaiaad6eacaWGSbWcdaWgaaqccasaaK qzadGaamyAaaqccasabaaaleqajugibiabggHiLdGaey4kaSIaamOu aSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWaiaadohaaaqcLb sacqGH9aqpcaaIWaaaaa@6537@  s = x, y                                      (14.a),

s={ x y    q s ={ Uh Vh ,    M l,i s ={ M l,i x M l,i y ,    R i s ={ R i x = A i g( h i H i k x + ( Uh ) i st ) R i y = A i g( h i H i k y + ( Vh ) i st )  st= n i 2 ( Uh ) i 2 + ( Vh ) i 2 h i 7/3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb GaaGjcVlabg2da9KqbaoaaceaakeaajugibuaabeqaceaaaOqaaKqz GeGaamiEaaGcbaqcLbsacaWG5baaaaGccaGL7baajugibiaabccacq GHshI3caqGGaGaamyCaKqbaoaaBaaajeaibaqcLbmacaWGZbaaleqa aKqzGeGaaGjcVlabg2da9iaayIW7juaGdaGabaGcbaqcLbsafaqabe GabaaakeaajugibiaadwfacaWGObaakeaajugibiaadAfacaWGObaa aaGccaGL7baajugibiaacYcacaqGGaGaaeiiaiaabccacaWGnbWcda qhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbmacaWG ZbaaaKqzGeGaaGjcVlabg2da9iaayIW7juaGdaGabaGcbaqcLbsafa qabeGabaaakeaajugibiaad2ealmaaDaaajeaibaqcLbmacaWGSbGa aiilaiaadMgaaKqaGeaajugWaiaadIhaaaaakeaajugibiaad2ealm aaDaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaKqaGeaajugWaiaa dMhaaaaaaaGccaGL7baajugibiaacYcacaqGGaGaaeiiaiaabccaca WGsbWcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaam4Caaaa jugibiaayIW7cqGH9aqpjuaGdaGabaGcbaqcLbsafaqabeGabaaake aajugibiaadkfalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbma caWG4baaaKqzGeGaaGjcVlabg2da9iaayIW7caWGbbqcfa4aaSbaaK qaGeaajugWaiaadMgaaSqabaqcLbsacaWGNbqcfa4aaeWaaOqaaKqz GeGaamiAaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqzGeGaaG jcVNqbaoaalaaakeaajugibiabgkGi2kaadIealmaaDaaajeaibaqc LbmacaWGPbaajeaibaqcLbmacaWGRbaaaaGcbaqcLbsacqGHciITca WG4baaaiabgUcaRKqbaoaabmaakeaajugibiaadwfacaWGObaakiaa wIcacaGLPaaajuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajugibi aayIW7caaMi8Uaam4CaiaadshaaOGaayjkaiaawMcaaaqaaKqzGeGa amOuaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWaiaadMhaaa qcLbsacaaMi8Uaeyypa0JaaGjcVlaadgeajuaGdaWgaaqcbasaaKqz adGaamyAaaWcbeaajugibiaadEgajuaGdaqadaGcbaqcLbsacaWGOb qcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaqcLbsacaaMi8Ecfa4a aSaaaOqaaKqzGeGaeyOaIyRaamisaSWaa0baaKqaGeaajugWaiaadM gaaKqaGeaajugWaiaadUgaaaaakeaajugibiabgkGi2kaadMhaaaGa ey4kaSscfa4aaeWaaOqaaKqzGeGaamOvaiaadIgaaOGaayjkaiaawM caaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqzGeGaam4Caiaa dshaaOGaayjkaiaawMcaaaaaaiaawUhaaKqzGeGaaeiiaiaadohaca WG0bGaeyypa0tcfa4aaSaaaOqaaKqzGeGaamOBaSWaa0baaKqaGeaa jugWaiaadMgaaKqaGeaajugWaiaaikdaaaqcLbsacaaMi8Ecfa4aaO aaaOqaaKqbaoaabmaakeaajugibiaadwfacaWGObaakiaawIcacaGL PaaalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaaIYaaaaK qzGeGaaGjcVlabgUcaRKqbaoaabmaakeaajugibiaadAfacaWGObaa kiaawIcacaGLPaaalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLb macaaIYaaaaaWcbeaaaOqaaKqzGeGaamiAaSWaa0baaKqaGeaajugW aiaadMgaaKqaGeaajugWaiaaiEdacaGGVaGaaG4maaaaaaaaaa@0E33@             (14.b),

where sub index i marks the variables in cell i, Fl,i and M l,i x( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgeaYxb9fr=xfr=x fr=BmeaabiqaciaacaGaaeqabaWaaeaaeaaakeaajugibiaad2ealm aaDaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaKqaGeaajugWaiaa dIhalmaabmaajeaibaqcLbmacaWG5baajeaicaGLOaGaayzkaaaaaa aa@4393@ , further specified, are the flux and the x(y) component of the momentum flux across side l (l = 1, ..., Nli) of cell i and M l,i x( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgeaYxb9fr=xfr=x fr=BmeaabiqaciaacaGaaeqabaWaaeaaeaaakeaajugibiaad2ealm aaDaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaKqaGeaajugWaiaa dIhalmaabmaajeaibaqcLbmacaWG5baajeaicaGLOaGaayzkaaaaaa aa@4393@  is the source term. The Manning coefficient in cell i, ni is computed as the mean value of the friction coefficients in the triangles sharing node i. The spatial gradients of the water level are kept constant in the prediction step of the hydrostatic problem (see Eq (14,b)), as motivated in [31,32].

In accordance to the formulation in Eqs. (A.3)-(A.5) in Appendix A, the linearized differential formulation of the DC problem is

h t + Uh x + Vh y = ( Uh ¯ ) x + ( Vh ¯ ) y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabiqaciaacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaa GcbaqcLbsacqGHciITcaWGObaakeaajugibiabgkGi2kaadshaaaGa ey4kaSscfa4aaSaaaOqaaKqzGeGaeyOaIyRaamyvaiaadIgaaOqaaK qzGeGaeyOaIyRaamiEaaaacqGHRaWkjuaGdaWcaaGcbaqcLbsacqGH ciITcaWGwbGaamiAaaGcbaqcLbsacqGHciITcaWG5baaaiabg2da9K qbaoaalaaakeaajugibiabgkGi2MqbaoaabmaakeaajuaGdaqdaaGc baqcLbsacaWGvbGaamiAaaaaaOGaayjkaiaawMcaaaqaaKqzGeGaey OaIyRaamiEaaaacqGHRaWkjuaGdaWcaaGcbaqcLbsacqGHciITjuaG daqadaGcbaqcfa4aa0aaaOqaaKqzGeGaamOvaiaadIgaaaaakiaawI cacaGLPaaaaeaajugibiabgkGi2kaadMhaaaaaaa@64BD@  (15),

q s t +g h ¯ H s +g n 2 ( q s ) ( ( Uh ) 2 + ( Vh ) 2 h 7/3 ) ¯ =g h ¯ H k s +g n 2 ( q ¯ s ) ( ( Uh ) 2 + ( Vh ) 2 h 7/3 ) ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabiqaciaacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaa GcbaqcLbsacqGHciITcaWGXbqcfa4aaSbaaKqaGeaajugWaiaadoha aSqabaaakeaajugibiabgkGi2kaadshaaaGaey4kaSIaam4zaiqadI gagaqeaKqbaoaalaaakeaajugibiabgkGi2kaadIeaaOqaaKqzGeGa eyOaIyRaam4CaaaacqGHRaWkcaWGNbGaaGPaVlaad6gajuaGdaahaa WcbeqcbasaaKqzadGaaGOmaaaajuaGdaqadaGcbaqcLbsacaWGXbqc fa4aaSbaaKqaGeaajugWaiaadohaaSqabaaakiaawIcacaGLPaaaju aGdaqdaaGcbaqcfa4aaeWaaOqaaKqbaoaalaaakeaajuaGdaGcaaGc baqcfa4aaeWaaOqaaKqzGeGaamyvaiaadIgaaOGaayjkaiaawMcaaK qbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey4kaSscfa4a aeWaaOqaaKqzGeGaamOvaiaadIgaaOGaayjkaiaawMcaaKqbaoaaCa aaleqajeaibaqcLbmacaaIYaaaaaWcbeaaaOqaaKqzGeGaamiAaKqb aoaaCaaaleqajeaibaqcLbmacaaI3aGaai4laiaaiodaaaaaaaGcca GLOaGaayzkaaaaaKqzGeGaeyypa0Jaam4zaiqadIgagaqeaKqbaoaa laaakeaajugibiabgkGi2kaadIeajuaGdaahaaWcbeqcbasaaKqzad Gaam4AaaaaaOqaaKqzGeGaeyOaIyRaam4CaaaacqGHRaWkcaWGNbGa aGPaVlaad6gajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajuaGda qadaGcbaqcLbsaceWGXbGbaebajuaGdaWgaaWcbaqcLbsacaWGZbaa leqaaaGccaGLOaGaayzkaaqcfa4aa0aaaOqaaKqbaoaabmaakeaaju aGdaWcaaGcbaqcfa4aaOaaaOqaaKqbaoaabmaakeaajugibiaadwfa caWGObaakiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaKqzadGaaG OmaaaajugibiabgUcaRKqbaoaabmaakeaajugibiaadAfacaWGObaa kiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaaaS qabaaakeaajugibiaadIgajuaGdaahaaWcbeqcbasaaKqzadGaaG4n aiaac+cacaaIZaaaaaaaaOGaayjkaiaawMcaaaaaaaa@A526@  (16),

and the symbol ( * ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceGGQa Gbaebaaaa@374A@ ) marks the mean-in-time value of the variable (*). The initial states of the CC and DC steps are the solutions of the CP and CC steps, respectively. In Eqs. (16) we assume negligible the difference between the sum of the convective terms and the mean-in-time values of the same quantities obtained after the (CP + CC) steps [31,32]. Symbols (*)k, (*)k+1/3, (*)k+2/3, ( ( * ˜ ) k+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaGGOa GabiOkayaaiaGaaiykaOWaaWbaaSqabeaacaWGRbGaey4kaSIaaGym aaaaaaa@3B5E@  mark the values of the variable (*) at the beginning of the time iteration, at the end of the convective prediction, convective correction and diffusive correction steps, respectively.

In the prediction steps, we assume, inside each computational cell, piecewise constant values of h, Uh, Vh. The correction system (15)-(16) is solved applying a technique similar to the linear (P1) conforming FE Galerkin scheme (see section 4.2). The procedure for the solution of the hydrostatic problem has been proposed in [32], where we adopted a cell-centered formulation with storage capacity concentrated in the mesh triangles. The node-centered formulation with respect to the Voronoi cells, presented in this paper, leads to a different spatial discretization of the governing equations. In the following sections 4.1 and 4.2, we only report the steps of the numerical formulation of the hydrostatic problem involved in the new spatial discretization. In test 3 (section 8.3) we compare the efficiency and accuracy of the two formulations using an analytical reference solution.

The prediction problem

All the computational cells, at the beginning of each time iteration, are ordered according to the value of the approximated potential ϕ i k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaaaa @3D06@ , known inside each cell, and computed according to the procedure proposed in [31,32]. This procedure, at the beginning of the time step, minimizes the differences [31,32]. From Eq. (12), we set the flux FLl,i through side l as

F L l.i =( n ^ i,ip q i ) d l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb GaamitaKqbaoaaBaaajeaibaqcLbmacaWGSbGaaiOlaiaadMgaaSqa baqcLbsacqGH9aqpjuaGdaqadaGcbaqcLbsaceWHUbGbaKaajuaGda WgaaqcbasaaKqzadGaamyAaiaacYcacaWGPbGaamiCaaWcbeaajugi biabgwSixlaahghajuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaaaO GaayjkaiaawMcaaKqzGeGaamizaKqbaoaaBaaajeaibaqcLbmacaWG SbGaaiilaiaadMgaaSqabaaaaa@5497@                               (17),

where n ^ i,ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiqah6 gagaqcaKqbaoaaBaaajeaibaqcLbmacaWGPbGaaiilaiaadMgacaWG Wbaaleqaaaaa@3D35@  is the unit vector parallel to side ri,ip (i.e., orthogonal to side l) positive outwards from i, i.e., oriented from i to ip, qi is the specific horizontal flow rate vector in cell i, qi = ((Uh)i, (Vh)i)T. and symbol "" MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqqFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaiOiaiabgw Sixlaackcaaaa@394B@  marks the dot product. Eq. (17) becomes

F L l,i =( ( Uh ) i ( x ip x i )+ ( Vh ) i ( y ip y i ) ) d l,i / | r i,ip | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb GaamitaKqbaoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqa baqcLbsacqGH9aqpjuaGdaqadaGcbaqcfa4aaeWaaOqaaKqzGeGaam yvaiaadIgaaOGaayjkaiaawMcaaKqbaoaaBaaajeaibaqcLbmacaWG PbaaleqaaKqbaoaabmaakeaajugibiaadIhajuaGdaWgaaqcbasaaK qzadGaamyAaiaadchaaSqabaqcLbsacqGHsislcaWG4bqcfa4aaSba aKqaGeaajugWaiaadMgaaSqabaaakiaawIcacaGLPaaajugibiabgU caRKqbaoaabmaakeaajugibiaadAfacaWGObaakiaawIcacaGLPaaa juaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajuaGdaqadaGcbaqcLb sacaWG5bqcfa4aaSbaaKqaGeaajugWaiaadMgacaWGWbaaleqaaKqz GeGaeyOeI0IaamyEaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaa GccaGLOaGaayzkaaaacaGLOaGaayzkaaqcfa4aaSGbaOqaaKqzGeGa amizaKqbaoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqaba aakeaajuaGdaabdaGcbaqcLbsacaWHYbqcfa4aaSbaaKqaGeaajugW aiaadMgacaGGSaGaamyAaiaadchaaSqabaaakiaawEa7caGLiWoaaa aaaa@7CE8@  (18).

If the GD property holds for side ri,ip, dl,i is positive (see Eq. (10) and Appendix B), and the sign of FLl,i only depends on the dot product ( n ^ i,ip q i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGabCOBayaajaqcfa4aaSbaaKqaGeaajugWaiaadMgacaGG SaGaamyAaiaadchaaSqabaqcLbsacqGHflY1caWHXbqcfa4aaSbaaK qaGeaajugWaiaadMgaaSqabaaakiaawIcacaGLPaaaaaa@461B@ , and a leaving (incoming) flux from (in) cell i is positive (negative). This condition is not fulfilled if the GD property is not satisfied, since dl,i is negative. The fluxes and momentum fluxes between cells i and ip in Eqs. (13)-(14) are defined as

F l,i =F L l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaajugi biabg2da9iaaysW7caWGgbGaamitaKqbaoaaBaaajeaibaqcLbmaca WGSbGaaiilaiaadMgaaSqabaaaaa@454E@  if ( F L l,i >0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb GaamitaKqbaoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqa baqcLbsacqGH+aGpcaaIWaaaaa@3F11@  and F L l,i >F L m,ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb GaamitaKqbaoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqa baqcLbsacqGH+aGpcaWGgbGaamitaKqbaoaaBaaajeaibaqcLbmaca WGTbGaaiilaiaadMgacaWGWbaaleqaaaaa@458A@ ) F l,i =F L m,ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaajugi biabg2da9iabgkHiTiaadAeacaWGmbqcfa4aaSbaaKqaGeaajugWai aad2gacaGGSaGaamyAaiaadchaaSqabaaaaa@45A4@  otherwise (19),

M l,i s = F l,i q s,i / h i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb WcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbma caWGZbaaaKqzGeGaaGzaVlabg2da9iaaxcW7caaMb8UaamOraSWaaS baaKqaGeaajugWaiaadYgacaGGSaGaamyAaaqcbasabaqcfa4aaSGb aOqaaKqzGeGaamyCaKqbaoaaBaaajeaibaqcLbmacaWGZbGaaiilai aadMgaaSqabaaakeaajugibiaadIgajuaGdaWgaaqcbasaaKqzadGa amyAaaWcbeaaaaaaaa@5453@  if F l,i =F L l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaajugi biaaygW7caaMc8Uaeyypa0JaaGzaVlaaykW7caWGgbGaamitaKqbao aaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqabaaaaa@49EB@  , M l,i s = F l,i q s,ip / h ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb WcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbma caWGZbaaaKqzGeGaaGzaVlabg2da9iaaygW7caaMe8UaamOraKqbao aaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqabaqcfa4aaSGb aOqaaKqzGeGaamyCaKqbaoaaBaaajeaibaqcLbmacaWGZbGaaiilai aadMgacaWGWbaaleqaaaGcbaqcLbsacaWGObqcfa4aaSbaaKqaGeaa jugWaiaadMgacaWGWbaaleqaaaaaaaa@56A6@  otherwise                         (20),

where m is the side of cell ip shared with cell i. Eqs. (19)-(20) guarantee the continuity of the flux and the momentum flux at each internal cell side, so that F l,i = F m,ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaajugi biabg2da9iabgkHiTiaadAeajuaGdaWgaaqcbasaaKqzadGaamyBai aacYcacaWGPbGaamiCaaWcbeaaaaa@44D3@  and M l,i x( y ) = M m,ip x( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb WcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbma caWG4bWcdaqadaqcbasaaKqzadGaamyEaaqcbaIaayjkaiaawMcaaa aajugibiabg2da9iaaysW7cqGHsislcaWGnbWcdaqhaaqcbasaaKqz adGaamyBaiaacYcacaWGPbGaamiCaaqcbasaaKqzadGaamiEaSWaae WaaKqaGeaajugWaiaadMhaaKqaGiaawIcacaGLPaaaaaaaaa@5226@  . For an external boundary side, the condition Fl,i = Fll,i holds for an outward oriented (positive) flux.

Thanks to the consecutive solution of the cells performed in the MAST procedure, the system of PDEs (13)-(14) for each cell i are written as a system of Ordinary Differential Equations (ODEs) (more details in [31-33]). After the cells ordering, in the CP step we proceed from the cell with the highest to the cell with the lowest value of ϕ k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2aaWbaaSqabeaacaWGRbaaaaaa@394B@ , and solve a ODEs system for each cell in the interval [ 0, Δt ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaqcLb saqaaaaaaaaaWdbiaaicdacaGGSaGaaeiiaiabfs5aejaadshaaOWd aiaawUfacaGLDbaaaaa@3D6C@  (from time level tk to time level tk+1/3and Δt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqqHuoarcaWG0baaaa@3954@  is the size of the time step). After that, in the CC step, we proceed from the cell with the lowest to the cell with the highest value of ϕ k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeqy1dy2aaWbaaSqabeaacaWGRbaaaaaa@394B@ , solving an ODEs system in each cell in the interval [ 0, Δt ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaqcLb saqaaaaaaaaaWdbiaaicdacaGGSaGaaeiiaiabfs5aejaadshaaOWd aiaawUfacaGLDbaaaaa@3D6C@ (from time level tk+1/3 to time level tk+2/3).

A 5th order Runge-Kutta method with adaptive stepsize control [42] is applied for the solution of the ODEs system. The ODEs system for the CP step is

d h i dt A i + 1 Δt Δt l=1,N l i δ l,i F l,i p,out ( t )dt = l=1,N l i ( 1 δ l,i ) F ¯ l,i p,in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadIgajuaGdaWgaaqcbasaaKqzadGaamyAaaWc beaaaOqaaKqzGeGaamizaiaadshaaaGaamyqaKqbaoaaBaaajeaiba qcLbmacaWGPbaaleqaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGeGa aGymaaGcbaqcLbsacqqHuoarcaWG0baaaKqbaoaapefakeaajuaGda aeqbGcbaqcLbsacqaH0oazjuaGdaWgaaqcbasaaKqzadGaamiBaiaa cYcacaWGPbaaleqaaKqzGeGaamOraKqbaoaaDaaajeaibaqcLbmaca WGSbGaaiilaiaadMgaaKqaGeaajugWaiaadchacaGGSaGaam4Baiaa dwhacaWG0baaaKqbaoaabmaakeaajugibiaadshaaOGaayjkaiaawM caaKqzGeGaamizaiaadshaaKqaGeaajugWaiaadYgacqGH9aqpcaaI XaGaaiilaiaad6eacaWGSbWcdaWgaaqccasaaKqzadGaamyAaaqcca sabaaaleqajugibiabggHiLdaajeaibaqcLbmacqqHuoarcaWG0baa leqajugibiabgUIiYdGaeyypa0tcfa4aaabuaOqaaKqbaoaabmaake aajugibiaaigdacqGHsislcqaH0oazjuaGdaWgaaqcbasaaKqzadGa amiBaiaacYcacaWGPbaaleqaaaGccaGLOaGaayzkaaqcLbsaceWGgb GbaebalmaaDaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaKqaGeaa jugWaiaadchacaGGSaGaamyAaiaad6gaaaaajeaibaqcLbmacaWGSb Gaeyypa0JaaGymaiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugW aiaadMgaaKGaGeqaaaWcbeqcLbsacqGHris5aaaa@974A@  (21),

d q s,i ( t ) dt A i + 1 Δt Δt R i p,s ( t )dt + 1 Δt Δt l=1,N l i δ l,i M l,i p,s,out ( t )dt = l=1,N l i ( 1 δ l,i ) M ¯ l,i p,s,in ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadghajuaGdaWgaaqcbasaaKqzadGaam4Caiaa cYcacaWGPbaaleqaaKqbaoaabmaakeaajugibiaadshaaOGaayjkai aawMcaaaqaaKqzGeGaamizaiaadshaaaGaamyqaKqbaoaaBaaajeai baqcLbmacaWGPbaaleqaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGe GaaGymaaqcaasaaKqzadGaeuiLdqKaamiDaaaajuaGdaWdrbGcbaqc LbsacaWGsbWcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaam iCaiaacYcacaWGZbaaaKqbaoaabmaakeaajugibiaadshaaOGaayjk aiaawMcaaKqzGeGaamizaiaadshaaKqaGeaajugWaiabfs5aejaads haaSqabKqzGeGaey4kIipacqGHRaWkjuaGdaWcaaGcbaqcLbsacaaI XaaajaaibaqcLbmacqqHuoarcaWG0baaaKqbaoaapefakeaajuaGda aeqbGcbaqcLbsacqaH0oazjuaGdaWgaaqcbasaaKqzadGaamiBaiaa cYcacaWGPbaaleqaaKqzGeGaamytaSWaa0baaKqaGeaajugWaiaadY gacaGGSaGaamyAaaqcbasaaKqzadGaamiCaiaacYcacaWGZbGaaiil aiaad+gacaWG1bGaamiDaaaajuaGdaqadaGcbaqcLbsacaWG0baaki aawIcacaGLPaaajugibiaadsgacaWG0baajeaibaqcLbmacaWGSbGa eyypa0JaaGymaiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugWai aadMgaaKGaGeqaaaWcbeqcLbsacqGHris5aaqcbasaaKqzadGaeuiL dqKaamiDaaWcbeqcLbsacqGHRiI8aiabg2da9Kqbaoaaqafakeaaju aGdaqadaGcbaqcLbsacaaIXaGaeyOeI0IaeqiTdqwcfa4aaSbaaKqa GeaajugWaiaadYgacaGGSaGaamyAaaWcbeaaaOGaayjkaiaawMcaaK qzGeGabmytayaaraWcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWG PbaajeaibaqcLbmacaWGWbGaaiilaiaadohacaGGSaGaamyAaiaad6 gaaaqcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaaajeai baqcLbmacaWGSbGaeyypa0JaaGymaiaacYcacaWGobGaamiBaSWaaS baaKGaGeaajugWaiaadMgaaKGaGeqaaaWcbeqcLbsacqGHris5aaaa @BF87@  s = x, y (22),

with R i x( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsb WcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaamiEaSWaaeWa aKqaGeaajugWaiaadMhaaKqaGiaawIcacaGLPaaaaaaaaa@4036@  defined in Eqs. (14,b), δ li =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKnaaBa aaleaacaWGSbGaamyAaaqabaGccqGH9aqpcaaIXaaaaa@3BC1@  if flux across side l is oriented outward i , 0 otherwise and

F l,i p,out =max( 0, F l,i ), M l,i p,s,out = F l,i p,out q s,i / h i ,s=x, y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb WcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbma caWGWbGaaiilaiaad+gacaWG1bGaamiDaaaajugibiabg2da9iGac2 gacaGGHbGaaiiEaKqbaoaabmaakeaajugibiaaicdacaGGSaGaamOr aSWaaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaqcbasabaaaki aawIcacaGLPaaajuaGcaGGSaGaaGPaVlaaykW7caaMc8EcLbsacaWG nbWcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLb macaWGWbGaaiilaiaadohacaGGSaGaam4BaiaadwhacaWG0baaaKqz GeGaeyypa0JaamOraSWaa0baaKqaGeaajugWaiaadYgacaGGSaGaam yAaaqcbasaaKqzadGaamiCaiaacYcacaWGVbGaamyDaiaadshaaaqc fa4aaSGbaOqaaKqzGeGaamyCaKqbaoaaBaaajeaibaqcLbmacaWGZb GaaiilaiaadMgaaSqabaaakeaajugibiaadIgajuaGdaWgaaqcbasa aKqzadGaamyAaaWcbeaaaaqcLbsacaGGSaGaaGzbVlaadohacqGH9a qpcaWG4bGaaiilaiaabccacaWG5baaaa@818D@ , if ϕ i k ϕ ip k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaKqz GeGaeyyzImRaeqy1dy2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaK qaGeaajugWaiaadUgaaaaaaa@46D3@  (23.a),

F l,i p,in =min( 0, F l,i ), M l,i p,x,in = F l,i p,in q s,ip / h ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb WcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbma caWGWbGaaiilaiaadMgacaWGUbaaaKqzGeGaeyypa0JaciyBaiaacM gacaGGUbqcfa4aaeWaaOqaaKqzGeGaaGimaiaacYcacaWGgbqcfa4a aSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaaaOGaayjkai aawMcaaKqzGeGaaiilaiaaykW7caaMc8UaaGPaVlaad2ealmaaDaaa jeaibaqcLbmacaWGSbGaaiilaiaadMgaaKqaGeaajugWaiaadchaca GGSaGaamiEaiaacYcacaWGPbGaamOBaaaajugibiabg2da9iaadAea lmaaDaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaKqaGeaajugWai aadchacaGGSaGaamyAaiaad6gaaaqcfa4aaSGbaOqaaKqzGeGaamyC aKqbaoaaBaaajeaibaqcLbmacaWGZbGaaiilaiaadMgacaWGWbaale qaaaGcbaqcLbsacaWGObqcfa4aaSbaaKqaGeaajugWaiaadMgacaWG Wbaaleqaaaaaaaa@7825@ , if ϕ i k < ϕ ip k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaKqz GeGaeyipaWJaeqy1dy2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaK qaGeaajugWaiaadUgaaaaaaa@4611@                                (23.b).

The right hand side (r.h.s.) of Eqs. (21)-(22) are the mean-in-time values of the incoming flux and x/y momentum fluxes, respectively, known after the solution of the linked ip cells with higher value of ϕ k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqy1dy McdaahaaWcbeqcbasaaKqzadGaam4Aaaaaaaa@3B1C@ , solved before i [31,32]. The gradients of H in the Rx(y) term are computed as proposed in 4.3.

Once the ODEs (21)-(22) are solved for cell i, the mean-in-time value of the total flux F ¯ i out MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaraWcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaam4Baiaa dwhacaWG0baaaaaa@3D26@ leaving from i, is given by the local mass balance [32],

F ¯ i out = F ¯ i in ( h i k+1/3 h i k )/ Δt A i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaamOraaaalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqc LbmacaWGVbGaamyDaiaadshaaaqcLbsacqGH9aqpjuaGdaqdaaGcba qcLbsacaWGgbaaaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugW aiaadMgacaWGUbaaaKqzGeGaeyOeI0scfa4aaSGbaOqaaKqbaoaabm aakeaajugibiaadIgalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqc LbmacaWGRbGaey4kaSIaaGymaiaac+cacaaIZaaaaKqzGeGaeyOeI0 IaamiAaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWaiaadUga aaaakiaawIcacaGLPaaaaeaajugibiabfs5aejaadshaaaGaamyqaK qbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaaaa@635E@  (24),

where F ¯ i in = l=1,N l i ( 1 δ l,i ) F ¯ l,i p,in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaamOraaaalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqc LbmacaWGPbGaamOBaaaajugibiabg2da9KqbaoaaqafakeaajuaGda qadaGcbaqcLbsacaaIXaGaeyOeI0IaeqiTdq2cdaWgaaqcbasaaKqz adGaamiBaiaacYcacaWGPbaajeaibeaaaOGaayjkaiaawMcaaKqzGe GabmOrayaaraWcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaa jeaibaqcLbmacaWGWbGaaiilaiaadMgacaWGUbaaaaqcbasaaKqzad GaamiBaiabg2da9iaaigdacaGGSaGaamOtaiaadYgalmaaBaaajiai baqcLbmacaWGPbaajiaibeaaaSqabKqzGeGaeyyeIuoaaaa@6039@  (i.e., the r.h.s. of Eq. (21)). Following [32], we compute the mean-in-time value of the flux F ¯ l,i out MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaraWcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqc LbmacaWGVbGaamyDaiaadshaaaaaaa@3EC7@ leaving from side l of i to the neighboring cell ip with ϕ i k ϕ ip k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaKqz GeGaeyyzImRaeqy1dy2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaK qaGeaajugWaiaadUgaaaaaaa@46D3@ , by partitioning ϕ i k ϕ ip k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaKqz GeGaeyyzImRaeqy1dy2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaK qaGeaajugWaiaadUgaaaaaaa@46D3@  on the base of the ratio between the flux F ¯ i out MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaraWcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaam4Baiaa dwhacaWG0baaaaaa@3D26@  and the sum of all the leaving fluxes to the neighboring cells at the end of the time step,

F ¯ l,i out = F ¯ i out ( F l,i p, out ) k+1/3 l=1,N l i δ l,i ( F l,i p, out ) k+1/3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaraqcfa4aa0baaKqaGeaajugWaiaadYgacaGGSaGaamyAaaqcbasa aKqzadGaam4BaiaadwhacaWG0baaaKqzGeGaeyypa0JabmOrayaara qcfa4aa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWaiaad+gacaWG 1bGaamiDaaaajuaGdaqadaGcbaqcLbsacaWGgbqcfa4aa0baaKqaGe aajugWaiaadYgacaGGSaGaamyAaaqcbasaaKqzadGaamiCaiaacYca caqGGaGaam4BaiaadwhacaWG0baaaaGccaGLOaGaayzkaaqcfa4aaW baaSqabKqaGeaajugWaiaadUgacqGHRaWkcaaIXaGaai4laiaaioda aaGcdaWcdaqaaaqaaaaajugibiabggHiLNqbaoaaBaaaleaajugWai aadYgacqGH9aqpcaaIXaGaaiilaiaad6eacaWGSbWcdaWgaaadbaqc LbmacaWGPbaameqaaaWcbeaajugibiabes7aKLqbaoaaBaaajeaiba qcLbmacaWGSbGaaiilaiaadMgaaSqabaqcfa4aaeWaaOqaaKqzGeGa amOraKqbaoaaDaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaKqaGe aajugWaiaadchacaGGSaGaaeiiaiaad+gacaWG1bGaamiDaaaaaOGa ayjkaiaawMcaaKqbaoaaCaaaleqajeaibaqcLbmacaWGRbGaey4kaS IaaGymaiaac+cacaaIZaaaaaaa@84FB@  (25.a),

and we estimate in a similar way the mean-in-time values of the leaving momentum flux M ¯ l,i x( y ),out MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabiqaciaacaGaaeqabaWaaeaaeaaakeaajugibiqad2 eagaqeaSWaa0baaKqaGeaajugWaiaadYgacaGGSaGaamyAaaqcbasa aKqzadGaamiEaSWaaeWaaKqaGeaajugWaiaadMhaaKqaGiaawIcaca GLPaaajugWaiaacYcacaWGVbGaamyDaiaadshaaaaaaa@4715@ . Finally, we set:

F ¯ m,ip in = F ¯ l,i out MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FasPYRqj0=yi0dg9Gr=dir=f0=yqaiVgFr0xfr=x fr=xgbpdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGgb GbaebalmaaDaaajeaibaqcLbmacaWGTbGaaiilaiaadMgacaWGWbaa jeaibaqcLbmacaWGPbGaamOBaaaajugibiabg2da9iqadAeagaqeaS Waa0baaKqaGeaajugWaiaadYgacaGGSaGaamyAaaqcbasaaKqzadGa am4BaiaadwhacaWG0baaaaaa@4CF6@   M ¯ m,ip s,in = M ¯ l,i s,out           s=x, y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FasPYRqj0=yi0dg9Gr=dir=f0=yqaiVgFr0xfr=x fr=xgbpdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGnb GbaebalmaaDaaajeaibaqcLbmacaWGTbGaaiilaiaadMgacaWGWbaa jeaibaqcLbmacaWGZbGaaiilaiaadMgacaWGUbaaaKqzGeGaeyypa0 JabmytayaaraWcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaa jeaibaqcLbmacaWGZbGaaiilaiaad+gacaWG1bGaamiDaaaacaqGGa qcLbsacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaWGZbGaeyypa0JaamiEaiaacYcacaqGGaGaamyEaa aa@5C8D@ (25.b),

for all the neighboring ip cells with ϕ i k ϕ ip k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaKqz GeGaeyyzImRaeqy1dy2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaK qaGeaajugWaiaadUgaaaaaaa@46D3@  and we proceed to solve system (21)-(22) for the ip cells among the unsolved ones. A similar algorithm is applied to solve the convective correction step, whose ODEs system for the generic cell i is

d h i dt A i + 1 Δt Δt l=1,N l i δ l,i F l,i c,out ( t )dt = l=1,N l i ( 1 δ l,i ) F ¯ l,i c,in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadIgajuaGdaWgaaWcbaqcLbsacaWGPbaaleqa aaGcbaqcLbsacaWGKbGaamiDaaaacaWGbbqcfa4aaSbaaKqaGeaaju gWaiaadMgaaSqabaqcLbsacqGHRaWkjuaGdaWcaaGcbaqcLbsacaaI Xaaakeaajugibiabfs5aejaadshaaaqcfa4aa8quaOqaaKqbaoaaqa fakeaajugibiabes7aKLqbaoaaBaaajeaibaqcLbmacaWGSbGaaiil aiaadMgaaSqabaqcLbsacaWGgbWcdaqhaaqcbasaaKqzadGaamiBai aacYcacaWGPbaajeaibaqcLbmacaWGJbGaaiilaiaad+gacaWG1bGa amiDaaaajuaGdaqadaGcbaqcLbsacaWG0baakiaawIcacaGLPaaaju gibiaadsgacaWG0baajeaibaqcLbmacaWGSbGaeyypa0JaaGymaiaa cYcacaWGobGaamiBaSWaaSbaaKGaGeaajugWaiaadMgaaKGaGeqaaa WcbeqcLbsacqGHris5aaWcbaqcLbsacqqHuoarcaWG0baaleqajugi biabgUIiYdGaeyypa0tcfa4aaabuaOqaaKqbaoaabmaakeaajugibi aaigdacqGHsislcqaH0oazjuaGdaWgaaqcbasaaKqzadGaamiBaiaa cYcacaWGPbaaleqaaaGccaGLOaGaayzkaaqcLbsaceWGgbGbaebalm aaDaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaKqaGeaajugWaiaa dogacaGGSaGaamyAaiaad6gaaaaajeaibaqcLbmacaWGSbGaeyypa0 JaaGymaiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugWaiaadMga aKGaGeqaaaWcbeqcLbsacqGHris5aaaa@9531@  (26),

d q s,i ( t ) dt A i + 1 Δt Δt l=1,N l i δ l,i M l,i c,s,out ( t )dt = l=1,N l i ( 1 δ l,i ) M ¯ l,i c,s,in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiaadghajuaGdaWgaaqcbasaaKqzadGaam4Caiaa cYcacaWGPbaaleqaaKqbaoaabmaakeaajugibiaadshaaOGaayjkai aawMcaaaqaaKqzGeGaamizaiaadshaaaGaamyqaKqbaoaaBaaajeai baqcLbmacaWGPbaaleqaaKqzGeGaey4kaSscfa4aaSaaaOqaaKqzGe GaaGymaaGcbaqcLbsacqqHuoarcaWG0baaaKqbaoaapefakeaajuaG daaeqbGcbaqcLbsacqaH0oazjuaGdaWgaaqcbasaaKqzadGaamiBai aacYcacaWGPbaaleqaaKqzGeGaamytaSWaa0baaKqaGeaajugWaiaa dYgacaGGSaGaamyAaaqcbasaaKqzadGaam4yaiaacYcacaWGZbGaai ilaiaad+gacaWG1bGaamiDaaaajuaGdaqadaGcbaqcLbsacaWG0baa kiaawIcacaGLPaaajugibiaadsgacaWG0baajeaibaqcLbmacaWGSb Gaeyypa0JaaGymaiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugW aiaadMgaaKGaGeqaaaWcbeqcLbsacqGHris5aaWcbaqcLbsacqqHuo arcaWG0baaleqajugibiabgUIiYdGaeyypa0tcfa4aaabuaOqaaKqb aoaabmaakeaajugibiaaigdacqGHsislcqaH0oazjuaGdaWgaaqcba saaKqzadGaamiBaiaacYcacaWGPbaaleqaaaGccaGLOaGaayzkaaqc LbsaceWGnbGbaebalmaaDaaajeaibaqcLbmacaWGSbGaaiilaiaadM gaaKqaGeaajugWaiaadogacaGGSaGaam4CaiaacYcacaWGPbGaamOB aaaaaKqaGeaajugWaiaadYgacqGH9aqpcaaIXaGaaiilaiaad6eaca WGSbWcdaWgaaqccasaaKqzadGaamyAaaqccasabaaaleqajugibiab ggHiLdaaaa@9EA7@  s = x, y (27),

with F l,i c,out =max( 0, F l,i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb WcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbma caWGJbGaaiilaiaad+gacaWG1bGaamiDaaaajugibiabg2da9iGac2 gacaGGHbGaaiiEaKqbaoaabmaakeaajugibiaaicdacaGGSaGaamOr aKqbaoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqabaaaki aawIcacaGLPaaaaaa@4F32@ M l,i c,s,out = F l,i c,out q s,i / h i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb WcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbma caWGJbGaaiilaiaadohacaGGSaGaam4BaiaadwhacaWG0baaaKqzGe Gaeyypa0JaamOraSWaa0baaKqaGeaajugWaiaadYgacaGGSaGaamyA aaqcbasaaKqzadGaam4yaiaacYcacaWGVbGaamyDaiaadshaaaqcfa 4aaSGbaOqaaKqzGeGaamyCaKqbaoaaBaaajeaibaqcLbmacaWGZbGa aiilaiaadMgaaSqabaaakeaajugibiaadIgajuaGdaWgaaqcbasaaK qzadGaamyAaaWcbeaaaaaaaa@5A94@   s = x, y if ϕ i k < ϕ ip k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaKqz GeGaeyipaWJaeqy1dy2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaK qaGeaajugWaiaadUgaaaaaaa@4611@         (28,a),

and F l,i c,in =min( 0, F l,i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aa0baaKqaGeaajugWaiaadYgacaGGSaGaamyAaaqcbasaaKqz adGaam4yaiaacYcacaWGPbGaamOBaaaajugibiabg2da9iGac2gaca GGPbGaaiOBaKqbaoaabmaakeaajugibiaaicdacaGGSaGaamOraKqb aoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqabaaakiaawI cacaGLPaaaaaa@4EAD@ M l,i c,s,in = F l,i c,in q s,ip / h ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb WcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbma caWGJbGaaiilaiaadohacaGGSaGaamyAaiaad6gaaaqcLbsacqGH9a qpcaWGgbWcdaqhaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaajeai baqcLbmacaWGJbGaaiilaiaadMgacaWGUbaaaKqbaoaalyaakeaaju gibiaadghajuaGdaWgaaqcbasaaKqzadGaam4CaiaacYcacaWGPbGa amiCaaWcbeaaaOqaaKqzGeGaamiAaKqbaoaaBaaajeaibaqcLbmaca WGPbGaamiCaaWcbeaaaaaaaa@5A72@  if ϕ i k ϕ ip k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaKqz GeGaeyyzImRaeqy1dy2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaK qaGeaajugWaiaadUgaaaaaaa@46D3@  (28,b).

The source terms is allocated in the convective prediction step [32]. The initial state in each cell of the convective correction step is the one computed in the same cell at the end of the convective prediction step. After the ODEs system (26)-(27) is solved in cell i, the mean-in-time value of the flux/momentum flux leaving from side l of i to the neighboring cell ip with ϕ i k < ϕ ip k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzlmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaKqz GeGaeyipaWJaeqy1dy2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaK qaGeaajugWaiaadUgaaaaaaa@4611@ , from tk+1/3 to tk+2/3, is computed similarly to the previous CP step. Further comments concerning the MAST procedure can be found in [31,32].

The mean-in-time values h ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmiAay aaraaaaa@3699@  and ( Uh ) 2 + ( Vh ) 2 h 7/3 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqbaoaanaaake aajuaGdaGcaaGcbaqcfa4aaeWaaOqaaKqzGeGaamyvaiaadIgaaOGa ayjkaiaawMcaaKqbaoaaCaaaleqajqwaa+FaaKqzadGaaGOmaaaaju gibiabgUcaRKqbaoaabmaakeaajugibiaadAfacaWGObaakiaawIca caGLPaaajuaGdaahaaWcbeqcKfaG=haajugWaiaaikdaaaGcdGafaU GaaeacuaiabGafacaajugibiacGa4GObqcfa4aiacGCaaaleqcGayc basaiacGjugWaiacGaiI3aGaiacGc+cacGaiaI4maaaaaSqabaaaaa aa@5640@ , required for the DC step (see Eqs. (16)), are obtained by numerical integration during the solution of each ODEs system during the convective prediction and correction steps [31,32].

The diffusive correction problem

After the prediction components of the water depths (levels) are computed with piecewise constant approximation (P0) inside each cell, a continuous piecewise linear (P1) shape is assigned to the water depths (levels) computed at the end of the (CP + CC) steps, obtained from the nodal values. If we move from the P0 to the P1 approximation, at a given time level, we do not alter the estimation of the global mass, if the polygons of the P0 approximation (Figure 2b) are the same as the ones of the P1 approximation. If we use, for the solution of the DC problem the conforming P1 Galerkin FE scheme, where the polygons are delimited by the centers of mass (see Appendix C), we introduce a small mass balance error. We avoid this error, using the same polygons for the P0 and P1 approximations. Other details in Appendix C.

We use a fully implicit time discretization of the DC problem (15)-(16) [32]. Under the hypothesis of fixed bed condition, we get the following differential equation for the generic cell i

H ˜ i k+1 H i k+2/3 Δt + ( U ˜ h ˜ ) k+1 x + ( V ˜ h ˜ ) k+1 y = ( Uh ) ¯ x + ( Vh ) ¯ y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGabmisayaaiaWcdaqhaaqcbasaaKqzadGaamyAaaqcbasa aKqzadGaam4AaiabgUcaRiaaigdaaaqcLbsacqGHsislcaWGibWcda qhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaam4AaiabgUcaRiaa ikdacaGGVaGaaG4maaaaaOqaaKqzGeGaeuiLdqKaamiDaaaacqGHRa WkjuaGdaWcaaGcbaqcLbsacqGHciITjuaGdaqadaGcbaqcLbsaceWG vbGbaGaaceWGObGbaGaaaOGaayjkaiaawMcaaKqbaoaaCaaaleqaje aibaqcLbmacaWGRbGaey4kaSIaaGymaaaaaOqaaKqzGeGaeyOaIyRa amiEaaaacqGHRaWkjuaGdaWcaaGcbaqcLbsacqGHciITjuaGdaqada GcbaqcLbsaceWGwbGbaGaaceWGObGbaGaaaOGaayjkaiaawMcaaKqb aoaaCaaaleqajeaibaqcLbmacaWGRbGaey4kaSIaaGymaaaaaOqaaK qzGeGaeyOaIyRaamyEaaaacqGH9aqpjuaGdaWcaaGcbaqcLbsacqGH ciITjuaGdaqdaaGcbaqcfa4aaeWaaOqaaKqzGeGaamyvaiaadIgaaO GaayjkaiaawMcaaaaaaeaajugibiabgkGi2kaadIhaaaGaey4kaSsc fa4aaSaaaOqaaKqzGeGaeyOaIyBcfa4aa0aaaOqaaKqbaoaabmaake aajugibiaadAfacaWGObaakiaawIcacaGLPaaaaaaabaqcLbsacqGH ciITcaWG5baaaaaa@823B@  (29),

q ˜ s,i k+1 q ˜ s,i k+2/3 Δt +g h ¯ i H ˜ k+1 s +g q ˜ s,i k+1 so r i =g h ¯ i H k s +g q ¯ s,i so r i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabiqaciaacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaa GcbaqcLbsaceWGXbGbaGaalmaaDaaajeaibaqcLbmacaWGZbGaaiil aiaadMgaaKqaGeaajugWaiaadUgacqGHRaWkcaaIXaaaaKqzGeGaaG jcVlabgkHiTiaayIW7ceWGXbGbaGaalmaaDaaajeaibaqcLbmacaWG ZbGaaiilaiaadMgaaKqaGeaajugWaiaadUgacqGHRaWklmaalyaaje aibaqcLbmacaaIYaaajeaibaqcLbmacaaIZaaaaaaaaOqaaKqzGeGa euiLdqKaamiDaaaacqGHRaWkcaWGNbGaaGPaVlqadIgagaqeaKqbao aaBaaajeaibaqcLbmacaWGPbaaleqaaKqbaoaalaaakeaajugibiab gkGi2kqadIeagaacaKqbaoaaCaaaleqajeaibaqcLbmacaWGRbGaey 4kaSIaaGymaaaaaOqaaKqzGeGaeyOaIyRaam4CaaaacaaMi8Uaey4k aSIaaGPaVlaayIW7caWGNbGaaGPaVlqadghagaacaSWaa0baaKqaGe aajugWaiaadohacaGGSaGaamyAaaqcbasaaKqzadGaam4AaiabgUca RiaaigdaaaqcLbsacaWGZbGaam4BaiaadkhajuaGdaWgaaqcbasaaK qzadGaamyAaaWcbeaajugibiabg2da9iaadEgacaaMc8UabmiAayaa raqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaqcfa4aaSaaaOqaaK qzGeGaeyOaIyRaamisaKqbaoaaCaaaleqajeaibaqcLbmacaWGRbaa aaGcbaqcLbsacqGHciITcaWGZbaaaiaayIW7cqGHRaWkcaWGNbGaaG PaVlqadghagaqeaKqbaoaaBaaajeaibaqcLbmacaWGZbGaaiilaiaa dMgaaSqabaqcLbsacaaMc8Uaam4Caiaad+gacaWGYbqcfa4aaSbaaK qaGeaajugWaiaadMgaaSqabaaaaa@A2BC@  s = x, y (30),

where so r i = n i 2 ( ( Uh ) i 2 + ( Vh ) i 2 / h i 7/3 ) ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabiqaciaacaGaaeqabaWaaeaaeaaakeaajugibiaado hacaWGVbGaamOCaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqz GeGaaGjcVlabg2da9iaayIW7caWGUbWcdaqhaaqcbasaaKqzadGaam yAaaqcbasaaKqzadGaaGOmaaaajuaGdaqdaaGcbaqcfa4aaeWaaOqa aKqbaoaalyaakeaajuaGdaGcaaGcbaqcfa4aaeWaaOqaaKqzGeGaam yvaiaadIgaaOGaayjkaiaawMcaaSWaa0baaKqaGeaajugWaiaadMga aKqaGeaajugWaiaaikdaaaqcLbsacaaMi8Uaey4kaSIaaGjcVNqbao aabmaakeaajugibiaadAfacaWGObaakiaawIcacaGLPaaalmaaDaaa jeaibaqcLbmacaWGPbaajeaibaqcLbmacaaIYaaaaaWcbeaaaOqaaK qzGeGaamiAaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWaiaa iEdacaGGVaGaaG4maaaaaaaakiaawIcacaGLPaaaaaaaaa@6957@ . From Eq. (30), the flow rate components at the end of the hydrostatic step are

q ˜ s,i k+1 = D ˜ i H ˜ k+1 s + k i s + q s,i k+2/3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGXb GbaGaalmaaDaaajeaibaqcLbmacaWGZbGaaiilaiaadMgaaKqaGeaa jugWaiaadUgacqGHRaWkcaaIXaaaaKqzGeGaeyypa0JaeyOeI0Iabm irayaaiaqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaqcfa4aaSaa aOqaaKqzGeGaeyOaIyRabmisayaaiaqcfa4aaWbaaSqabKqaGeaaju gWaiaadUgacqGHRaWkcaaIXaaaaaGcbaqcLbsacqGHciITcaWGZbaa aiabgUcaRiaadUgalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLb macaWGZbaaaKqzGeGaey4kaSIaamyCaSWaa0baaKqaGeaajugWaiaa dohacaGGSaGaamyAaaqcbasaaKqzadGaam4AaiabgUcaRiaaikdaca GGVaGaaG4maaaaaaa@635F@  s = x, y (31),

with coefficients D ˜ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGeb GbaGaajuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaaaaa@3A5B@  and k i x( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb WcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaamiEaSWaaeWa aKqaGeaajugWaiaadMhaaKqaGiaawIcacaGLPaaaaaaaaa@404F@  defined as

D ˜ i ={ g h ¯ i Δt/ [ 1+Δtgso r i ]      if    ϕ i k ϕ ip k g h ¯ ip Δt/ [ 1+Δtgso r ip ]  if    ϕ ip k > ϕ i k             k i s = D ˜ i H k s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGeb GbaGaajuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajugibiabg2da 9KqbaoaaceaajugibqaabeGcbaqcfa4aaSGbaOqaaKqzGeGaam4zai qadIgagaqeaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqzGeGa euiLdqKaamiDaaGcbaqcfa4aamWaaOqaaKqzGeGaaGymaiabgUcaRi abfs5aejaadshacaaMc8Uaam4zaiaaykW7caaMc8Uaam4Caiaad+ga caWGYbqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaaakiaawUfaca GLDbaaaaqcLbsacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabMga caqGMbGaaeiiaiaabccacaqGGaGaeqy1dy2cdaqhaaqcbasaaKqzad GaamyAaaqcbasaaKqzadGaam4AaaaajugibiabgwMiZkabew9aMTWa a0baaKqaGeaajugWaiaadMgacaWGWbaajeaibaqcLbmacaWGRbaaaa Gcbaqcfa4aaSGbaOqaaKqzGeGaam4zaiqadIgagaqeaKqbaoaaBaaa jeaibaqcLbmacaWGPbGaamiCaaWcbeaajugibiabfs5aejaadshaaO qaaKqbaoaadmaakeaajugibiaaigdacqGHRaWkcqqHuoarcaWG0bGa aGPaVlaadEgacaaMc8UaaGPaVlaadohacaWGVbGaamOCaKqbaoaaBa aajeaibaqcLbmacaWGPbGaamiCaaWcbeaaaOGaay5waiaaw2faaaaa jugibiaabccacaqGPbGaaeOzaiaabccacaqGGaGaaeiiaiabew9aML qbaoaaDaaajeaibaqcLbmacaWGPbGaamiCaaqcbasaaKqzadGaam4A aaaajugibiabg6da+iabew9aMLqbaoaaDaaajeaibaqcLbmacaWGPb aajeaibaqcLbmacaWGRbaaaaaakiaawUhaaKqzGeGaaGPaVlaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaam4AaKqbaoaaDaaajeaibaqcLbmacaWGPbaajeai baqcLbmacaWGZbaaaKqzGeGaeyypa0Jabmirayaaiaqcfa4aaSbaaK qaGeaajugWaiaadMgaaSqabaqcfa4aaSaaaOqaaKqzGeGaeyOaIyRa amisaKqbaoaaCaaaleqajeaibaqcLbmacaWGRbaaaaGcbaqcLbsacq GHciITcaWGZbaaaaaa@C316@  s = x, y (32).

We merge Eqs. (31) in Eq. (29) and, after space integration and application of the Green's theorem, we get the following balance law for cell i

A i H i t d A i l=1,N l i ( D ˜ i H ˜ k+1 n i,ip d l,i ) = l=1,N l i ( D ˜ i H ˜ k n i,ip d l,i ) + l=1,N l i ( q ¯ i q i k+2/3 ) n ^ i,ip d l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqVepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaajuaGdaWdrb Gcbaqcfa4aaSaaaOqaaKqzGeGaeyOaIyRaamisaKqbaoaaBaaajeai baqcLbmacaWGPbaaleqaaaGcbaqcLbsacqGHciITcaWG0baaaiaads gacaWGbbqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaaajeaibaqc LbmacaWGbbWcdaWgaaqccasaaKqzadGaamyAaaqccasabaaaleqaju gibiabgUIiYdGaeyOeI0scfa4aaabuaOqaaKqbaoaabmaakeaajugi biqadseagaacaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqbao aalaaakeaajugibiabgkGi2kqadIeagaacaKqbaoaaCaaaleqajeai baqcLbmacaWGRbGaey4kaSIaaGymaaaaaOqaaKqzGeGaeyOaIyRaam OBaKqbaoaaBaaajeaibaqcLbmacaWGPbGaaiilaiaadMgacaWGWbaa leqaaaaajugibiaadsgajuaGdaWgaaqcbasaaKqzadGaamiBaiaacY cacaWGPbaaleqaaaGccaGLOaGaayzkaaaajeaibaqcLbmacaWGSbGa eyypa0JaaGymaiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugWai aadMgaaKGaGeqaaaWcbeqcLbsacqGHris5aiabg2da9iabgkHiTKqb aoaaqafakeaajuaGdaqadaGcbaqcLbsaceWGebGbaGaajuaGdaWgaa qcbasaaKqzadGaamyAaaWcbeaajuaGdaWcaaGcbaqcLbsacqGHciIT ceWGibGbaGaajuaGdaahaaWcbeqcbasaaKqzadGaam4AaaaaaOqaaK qzGeGaeyOaIyRaamOBaKqbaoaaBaaajeaibaqcLbmacaWGPbGaaiil aiaadMgacaWGWbaaleqaaaaajugibiaadsgajuaGdaWgaaqcbasaaK qzadGaamiBaiaacYcacaWGPbaaleqaaaGccaGLOaGaayzkaaaajeai baqcLbmacaWGSbGaeyypa0JaaGymaiaacYcacaWGobGaamiBaSWaaS baaKGaGeaajugWaiaadMgaaKGaGeqaaaWcbeqcLbsacqGHris5aiab gUcaRKqbaoaaqafakeaajuaGdaqadaGcbaqcLbsaceWHXbGbaebaju aGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajugibiabgkHiTiaahgha lmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbGaey4kaS IaaGOmaiaac+cacaaIZaaaaaGccaGLOaGaayzkaaqcLbsacqGHflY1 ceWHUbGbaKaajuaGdaWgaaqcbasaaKqzadGaamyAaiaacYcacaWGPb GaamiCaaWcbeaajugibiaadsgajuaGdaWgaaqcbasaaKqzadGaamiB aiaacYcacaWGPbaaleqaaaqcbasaaKqzadGaamiBaiabg2da9iaaig dacaGGSaGaamOtaiaadYgalmaaBaaajiaibaqcLbmacaWGPbaajiai beaaaSqabKqzGeGaeyyeIuoaaaa@CEE3@  (33),

where H/ n i,ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaamaalyaabaqcLb sacqGHciITcaWGibaakeaajugibiabgkGi2kaad6gajuaGdaWgaaqc basaaKqzadGaamyAaiaacYcacaWGPbGaamiCaaWcbeaaaaaaaa@4061@  is the water level gradient along the n ^ i,ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWHUb GbaKaajuaGdaWgaaqcbasaaKqzadGaamyAaiaacYcacaWGPbGaamiC aaWcbeaaaaa@3D1D@  direction and the other symbols are specified as above. Concerning the second term on the r.h.s. of Eq. (33), we have

l=1,N l i ( q ¯ i n ^ i,ip d l,i ) = F ¯ i in F ¯ i out = H i k+2/3 H i k Δt A i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqbaoaaqafake aajuaGdaqadaGcbaqcLbsacaaMc8UabCyCayaaraWcdaWgaaqcbasa aKqzadGaamyAaaqcbasabaqcLbsacqGHflY1ceWHUbGbaKaajuaGda WgaaqcbasaaKqzadGaamyAaiaacYcacaWGPbGaamiCaaWcbeaajugi biaadsgajuaGdaWgaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaale qaaaGccaGLOaGaayzkaaaajeaibaqcLbmacaWGSbGaeyypa0JaaGym aiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugWaiaadMgaaKGaGe qaaaWcbeqcLbsacqGHris5aiabg2da9iqadAeagaqeaSWaa0baaKqa GeaajugWaiaadMgaaKqaGeaajugWaiaadMgacaWGUbaaaKqzGeGaey OeI0IabmOrayaaraWcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqz adGaam4BaiaadwhacaWG0baaaOGaeyypa0ZaaSaaaeaajugibiaadI ealmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbGaey4k aSIaaGOmaiaac+cacaaIZaaaaKqzGeGaeyOeI0IaamisaSWaa0baaK qaGeaajugWaiaadMgaaKqaGeaajugWaiaadUgaaaaakeaacqqHuoar caWG0baaaiaadgeadaWgaaWcbaGaamyAaaqabaaaaa@7F14@  (34),

while q i k+2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqqFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWHXb WcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaam4AaiabgUca RiaaikdacaGGVaGaaG4maaaaaaa@3F07@  is computed after the (CP + CC) steps. After discretization in time, Eq. (33) becomes

H ˜ i k+1 H i k+2/3 Δt A i + l=1,N l i F ˜ l,i = l=1,N l i b ˜ l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaamaalaaajaaiba GabmisayaaiaGcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGa am4AaiabgUcaRiaaigdaaaqcaaIaeyOeI0IaamisaSWaa0baaKqaGe aajugWaiaadMgaaKqaGeaajugWaiaadUgacqGHRaWkcaaIYaGaai4l aiaaiodaaaaajaaibaGaeuiLdqKaamiDaaaajugibiaadgealmaaBa aajeaibaqcLbmacaWGPbaajeaibeaajugibiabgUcaRKqbaoaaqafa keaajugibiqadAeagaacaKqbaoaaBaaajeaibaqcLbmacaWGSbGaai ilaiaadMgaaSqabaaajeaibaqcLbmacaWGSbGaeyypa0JaaGymaiaa cYcacaWGobGaamiBaSWaaSbaaKGaGeaajugWaiaadMgaaKGaGeqaaa WcbeqcLbsacqGHris5aiabg2da9Kqbaoaaqafakeaajugibiqadkga gaacaKqbaoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqaba aajeaibaqcLbmacaWGSbGaeyypa0JaaGymaiaacYcacaWGobGaamiB aSWaaSbaaKGaGeaajugWaiaadMgaaKGaGeqaaaWcbeqcLbsacqGHri s5aaaa@73EE@  i = 1, ..., N (35),

where F ˜ l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgea0lrP0xe9Fve9 Fve9VXqaaeGabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3E66@  is the corrective flux across side l of cell i

F ˜ l,i = D ˜ i H ˜ k+1 n i,ip d l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa jugibiabg2da9iabgkHiTiqadseagaacaKqbaoaaBaaajeaibaqcLb macaWGPbaaleqaaOWaaSaaaeaajugibiabgkGi2kqadIeagaacaKqb aoaaCaaaleqajeaibaqcLbmacaWGRbGaey4kaSIaaGymaaaaaOqaaK qzGeGaeyOaIyRaamOBaKqbaoaaBaaajeaibaqcLbmacaWGPbGaaiil aiaadMgacaWGWbaaleqaaaaajugibiaadsgajuaGdaWgaaqcKfaG=h aajugWaiaadYgacaGGSaGaamyAaaqcbawabaaaaa@59B3@                                                                            (36.a),

and the term b ˜ l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgea0lrP0xe9Fve9 Fve9VXqaaeGabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOyay aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3E82@  on the r.h.s. of Eq. (35) is

b ˜ l,i = D ˜ i H ˜ k n i,ip d l,i +( q ¯ i q i k+2/3 ) n ^ i,ip i,ip d l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOyay aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa jugibiabg2da9iabgkHiTiqadseagaacaKqbaoaaBaaajeaibaqcLb macaWGPbaaleqaaOWaaSaaaeaajugibiabgkGi2kqadIeagaacaKqb aoaaCaaaleqajeaibaqcLbmacaWGRbaaaaGcbaqcLbsacqGHciITca WGUbqcfa4aaSbaaKqaGeaajugWaiaadMgacaGGSaGaamyAaiaadcha aSqabaaaaKqzGeGaamizaKqbaoaaBaaajeaibaqcLbmacaWGSbGaai ilaiaadMgaaSqabaqcLbsacqGHRaWkjuaGdaqadaGcbaqcLbsaceWH XbGbaebajuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajugibiabgk HiTiaahghalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWG RbGaey4kaSIaaGOmaiaac+cacaaIZaaaaaGccaGLOaGaayzkaaqcLb sacqGHflY1ceWHUbGbaKaajuaGdaWgaaqcbasaaKqzadGaamyAaiaa cYcacaWGPbGaamiCaaWcbeaajuaGdaWgaaWcbaqcLbsacaWGPbGaai ilaiaadMgacaWGWbaaleqaaKqzGeGaamizaKqbaoaaBaaajeaibaqc LbmacaWGSbGaaiilaiaadMgaaSqabaaaaa@7CED@  (36.b).

Due to the P1 spatial approximation, H ˜ k+1 / n i,ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaamaalyaabaqcLb sacqGHciITceWGibGbaGaajuaGdaahaaWcbeqcbasaaKqzadGaam4A aiabgUcaRiaaigdaaaaakeaajugibiabgkGi2kaad6gajuaGdaWgaa qcbasaaKqzadGaamyAaiaacYcacaWGPbGaamiCaaWcbeaaaaaaaa@4510@  only depends on the values at nodes i and ip (since n ^ i,ip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiqah6 gagaqcaKqbaoaaBaaajeaibaqcLbmacaWGPbGaaiilaiaadMgacaWG WbaaleqaaKqzGeGaaGjcVdaa@3F55@ is parallel to ri,ip and orthogonal to side l, see sections 3.2 and 4.1), and F ˜ l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgea0lrP0xe9Fve9 Fve9VXqaaeGabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3E66@  is written

F ˜ l,i = D ˜ i ( H ˜ ip k+1 H ˜ i k+1 ) | r i,ip | d l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa jugibiabg2da9iabgkHiTiqadseagaacaKqbaoaaBaaajeaibaqcLb macaWGPbaaleqaaKqbaoaabmaakeaajugibiqadIeagaacaSWaa0ba aKqaGeaajugWaiaadMgacaWGWbaajeaibaqcLbmacaWGRbGaey4kaS IaaGymaaaajugibiabgkHiTiqadIeagaacaSWaa0baaKqaGeaajugW aiaadMgaaKqaGeaajugWaiaadUgacqGHRaWkcaaIXaaaaaGccaGLOa GaayzkaaWaaSGaaeaaaeaaaaqcLbsacaaMe8Ecfa4aaqWaaOqaaKqz GeGaaCOCaKqbaoaaBaaajeaibaqcLbmacaWGPbGaaiilaiaadMgaca WGWbaaleqaaaGccaGLhWUaayjcSdqcLbsacaWGKbqcfa4aaSbaaKqa GeaajugWaiaadYgacaGGSaGaamyAaaWcbeaaaaa@6823@  (37).

Eqs. (35) constitute a linear system of order N in the Hi (i = 1, …, N) unknowns, whose matrix is sparse and symmetric. Generally the non-zero entries are about 5% or less of the total number of matrix coefficients. The diagonal and off-diagonal matrix coefficients are, respectively

s i,i = A i Δt + l=1,N l i D ˜ i | r i,ip | d l,i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGaam4CaK qbaoaaBaaajeaibaqcLbmacaWGPbGaaiilaiaadMgaaSqabaGccqGH 9aqpdaWcaaqaaKqzGeGaamyqaKqbaoaaBaaajeaibaqcLbmacaWGPb aaleqaaaGcbaqcLbsacqqHuoarcaWG0baaaOGaey4kaSYaaabuaeaa daWcaaqaaKqzGeGabmirayaaiaqcfa4aaSbaaKqaGeaajugWaiaadM gaaSqabaaakeaajuaGdaabdaGcbaqcLbsacaWHYbqcfa4aaSbaaKqa GeaajugWaiaadMgacaGGSaGaamyAaiaadchaaSqabaaakiaawEa7ca GLiWoaaaqcLbsacaWGKbqcfa4aaSbaaKqaGeaajugWaiaadYgacaGG SaGaamyAaaWcbeaaaKqaGeaajugWaiaadYgacqGH9aqpcaaIXaGaai ilaiaad6eacaWGSbWcdaWgaaqccasaaKqzadGaamyAaaqccasabaaa leqaniabggHiLdGccaaMc8oaaa@66CF@ s i,ip = D ˜ i | r i,ip | d l,i δ i,ip MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGaam4CaK qbaoaaBaaajeaibaqcLbsacaWGPbGaaiilaiaadMgacaWGWbaaleqa aKqzGeGaeyypa0JaeyOeI0IcdaWcaaqcfayaaiqadseagaacamaaBa aajuaibaGaamyAaaqcfayabaaabaWaaqWaaeaacaWHYbWaaSbaaKqb GeaacaWGPbGaaiilaiaadMgacaWGWbaajuaGbeaaaiaawEa7caGLiW oaaaqcLbwacaaMc8EcLbsacaaMi8UaamizaKqbaoaaBaaajeaibaqc LbmacaWGSbGaaiilaiaadMgaaSqabaqcLbsacaaMc8UaeqiTdqwcfa 4aaSbaaKqaGeaajugWaiaadMgacaGGSaGaamyAaiaadchaaSqabaaa aa@5D1C@  i, ip = 1, ..., N (38),

where δ i ,ip = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH0oazl8aadaWgaaqcbasaaKqzadWdbiaadMgaaKqaG8aa beaalmaaBaaajeaibaqcLbmapeGaaiilaiaadMgacaWGWbaajeaipa qabaGcpeGaeyypa0Jaaeiiaiaaigdaaaa@4242@  if nodes i and ip are linked, otherwise it is equal to zero.

According to Eqs. (37)-(38), the off-diagonal coefficients are non-negative if the GD mesh property is not satisfied for side ri,ip and the M-matrix property is lost. In this case, dl,i is negative (see Eq. (10) and Appendix B) and the sign of F ˜ l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3B0F@  is not consistent with the difference ( H ˜ ip k+1 H ˜ i k+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiqadIeagaacaSWaa0baaKqaGeaajugWaiaadMgacaWGWbaa jeaibaqcLbmacaWGRbGaey4kaSIaaGymaaaajugibiabgkHiTiqadI eagaacaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWaiaadUga cqGHRaWkcaaIXaaaaaGccaGLOaGaayzkaaaaaa@4898@  in Eq. (37). On the opposite, if side ri,ip satisfies the GD property, dl,i is positive, and the sign of the flux in Eq. (37) only depends on the difference of the water levels in cells i and ip. If the GD property is satisfied for all the sides of the mesh, the matrix of system (35) is a M-matrix, positive-definite and, according to [41], the system is well-conditioned.

The numerical scheme adopted for the solution of the DC problem has a structure similar to the P1 conforming FE Galerkin scheme, and we present and discuss analogies and differences between the two numerical formulations in Appendix C.

We solve system (35) by a preconditioned conjugate gradient method (with incomplete Cholesky preconditioning). Due to the symmetry of the matrix of system (35), we only save the non-zero entries of the lower triangular matrix, using a Symmetric Coordinate Storage algorithm [43 ].

After the solution of system (35) for the H unknowns, we compute the gradients of the water level x( y ) H ˜ i k+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrpepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiabgE GirNqbaoaaBaaaleaajugibiaadIhajuaGdaqadaWcbaqcLbsacaWG 5baaliaawIcacaGLPaaaaeqaaKqzGeGabmisayaaiaWcdaqhaaqcba saaKqzadGaamyAaaqcbasaaKqzadGaam4AaiabgUcaRiaaigdaaaaa aa@467F@  at the end of the DC step as shown in the next section. As specified in section 4.1, the values of the gradients of the water level are assumed constant during the solution of the two convective steps of the hydrostatic problem in the next time iteration [31,32]. Finally, the components of the flow rate q ˜ s,i k+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGXb GbaGaalmaaDaaajeaibaqcLbmacaWGZbGaaiilaiaadMgaaKqaGeaa jugWaiaadUgacqGHRaWkcaaIXaaaaaaa@3F89@  (s = x, y) are updated according to Eqs. (31)-(32).

According to Eq. (37) and the definition of the coefficient D ˜ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGeb GbaGaajuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaaaaa@3A5B@  in (Eq. (32)), the flux F ˜ l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgea0lrP0xe9Fve9 Fve9VXqaaeGabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3E66@  leaving from cell i (ip) to ip (i) is equal to the flux entering cell ip (i) from cell i (ip). If F ˜ l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgea0lrP0xe9Fve9 Fve9VXqaaeGabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3E66@  is leaving from i to ip or vice versa depends on the difference ( H ˜ i k+1 H ˜ ip k+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiqadIeagaacaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaa jugWaiaadUgacqGHRaWkcaaIXaaaaKqzGeGaeyOeI0Iabmisayaaia WcdaqhaaqcbasaaKqzadGaamyAaiaadchaaKqaGeaajugWaiaadUga cqGHRaWkcaaIXaaaaaGccaGLOaGaayzkaaaaaa@4898@  (see Eq. (37).

Computation of the spatial gradients of the water level

We compute the gradient of H in cell i as follows. Call n ^ q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWHUb GbaKaajuaGdaWgaaqcbasaaKqzadGaamyCaaWcbeaaaaa@3A3E@  the unit vector parallel to the flow vector in cell i. Let H i,ip MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqGHhi s0caWGibqcfa4aaSbaaKqaGeaajugWaiaadMgacaGGSaGaamyAaiaa dchaaSqabaaaaa@3E15@  be the vector computed as (ip is one of the cells linked with i),

H i,ip = ( H i H ip )/ r i,ip MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqGHhi s0caWGibqcfa4aaSbaaKqaGeaajugWaiaadMgacaGGSaGaamyAaiaa dchaaSqabaqcLbsacqGH9aqpjuaGdaWcgaGcbaqcfa4aaeWaaOqaaK qzGeGaamisaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqzGeGa eyOeI0IaamisaKqbaoaaBaaajeaibaqcLbmacaWGPbGaamiCaaWcbe aaaOGaayjkaiaawMcaaaqaaKqzGeGaaCOCaKqbaoaaBaaajeaibaqc LbmacaWGPbGaaiilaiaadMgacaWGWbaaleqaaaaaaaa@543A@  (39).

We assume

H i = ip=1,N l i max( 0,( n ^ i,ip n ^ q ) H i,ip ) ip=1,N l i max( 0,( n ^ i,ip n ^ q ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqGHhi s0caWGibqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcfa4aaabeaOqaaKqzGeGaciyBaiaacggaca GG4bqcfa4aaeWaaOqaaKqzGeGaaGimaiaacYcajuaGdaqadaGcbaqc LbsaceWHUbGbaKaajuaGdaWgaaqcbasaaKqzadGaamyAaiaacYcaca WGPbGaamiCaaWcbeaajugibiabgwSixlqah6gagaqcaKqbaoaaBaaa jeaibaqcLbmacaWGXbaaleqaaaGccaGLOaGaayzkaaqcLbsacqGHhi s0caWGibqcfa4aaSbaaKqaGeaajugWaiaadMgacaGGSaGaamyAaiaa dchaaSqabaaakiaawIcacaGLPaaaaKqaGeaajugWaiaadMgacaWGWb Gaeyypa0JaaGymaiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugW aiaadMgaaKGaGeqaaaWcbeqcLbsacqGHris5aaGcbaqcfa4aaabeaO qaaKqzGeGaciyBaiaacggacaGG4bqcfa4aaeWaaOqaaKqzGeGaaGim aiaacYcajuaGdaqadaGcbaqcLbsaceWHUbGbaKaajuaGdaWgaaqcba saaKqzadGaamyAaiaacYcacaWGPbGaamiCaaWcbeaajugibiabgwSi xlqah6gagaqcaKqbaoaaBaaajeaqbaqcLboacaWGXbaaleqaaaGcca GLOaGaayzkaaaacaGLOaGaayzkaaaajeaibaqcLbmacaWGPbGaamiC aiabg2da9iaaigdacaGGSaGaamOtaiaadYgalmaaBaaajiaibaqcLb macaWGPbaajiaibeaaaSqabKqzGeGaeyyeIuoaaaaaaa@9010@  (40),

where n ^ i,ip n ^ q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWHUb GbaKaajuaGdaWgaaqcbasaaKqzadGaamyAaiaacYcacaWGPbGaamiC aaWcbeaajugibiabgwSixlqah6gagaqcaKqbaoaaBaaajeaibaqcLb macaWGXbaaleqaaaaa@43B1@  is the dot product between the direction of side ri,ip and the unitary flow rate vector. If cell i lies on an impervious boundary, the flow rate vector is parallel to the boundary and H i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqGHhi s0caWGibqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaaaaa@3B82@ , computed as in Eq. (40), is projected along the boundary direction.

The expression in Eq. (40) can be regarded as an upwind formulation, where H i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqGHhi s0caWGibqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaaaaa@3B82@  depends on the differences between Hi and the values of H in the downstream (along the flow direction) ip cells. This is consistent with the procedure adopted in the (CP + CC) steps, where the unknown fluxes/momentum fluxes are the ones leaving from i to the downstream ip cells.

The Non-Hydrostatic Problem

The divergence-free continuity Eq. (1) is integrated to the control volume τ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHepaDk8aadaWgaaqcbasaaKqzadWdbiaadMgaaSWdaeqa aaaa@3B64@ corresponding to the computational cell i, τ i = ( A i × h ˜ i k + 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiXdq3damaaBaaaleaapeGaamyAaaWdaeqaaOWdbiabg2da98aa caGGOaWdbiaadgeapaWaaSbaaSqaa8qacaWGPbaapaqabaGcpeGaey 41aq7daiqadIgagaacamaaDaaaleaacaWGPbaabaGaam4AaiabgUca RiaaigdaaaGccaGGPaaaaa@4507@ and, due to the divergence theorem, we obtain the following balance relation between the horizontal and the vertical fluxes,

Σ i ( U h n ^ Σ ) d σ + A i ( w s , i w b , i ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aa8GeaO qaaKqbaoaabmaakeaajugibiaahwfajuaGdaWgaaqcbasaaKqzadGa amiAaaWcbeaajugibiabgwSixlqah6gagaqcaKqbaoaaBaaaleaaju gibiabfo6atbWcbeaaaOGaayjkaiaawMcaaKqzGeGaaGjcVlaayIW7 caWGKbGaeq4Wdmhaleaajugibiabfo6atLqbaoaaBaaajiaibaqcLb macaWGPbaameqaaaWcbeqcLbsacqGHRiI8cqGHRiI8aiabgUcaRiaa dgeajuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajuaGdaqadaGcba qcLbsacaWG3bqcfa4aaSbaaKqaGeaajugWaiaadohacaGGSaGaamyA aaWcbeaajugibiabgkHiTiaadEhajuaGdaWgaaqcbasaaKqzadGaam OyaiaacYcacaWGPbaaleqaaaGccaGLOaGaayzkaaqcLbsacqGH9aqp caaIWaaaaa@6A14@ (41.a),

where Σ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqqHJo WujuaGdaWgaaqccasaaKqzadGaamyAaaadbeaaaaa@3B09@ is the vertical surface of τ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHepaDk8aadaWgaaqcbasaaKqzadWdbiaadMgaaSWdaeqa aaaa@3B64@ ( i h ˜ i k + 1 × Γ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabggHiLpaaBa aaleaacaWGPbaabeaakiqadIgagaacamaaDaaaleaacaWGPbaabaGa am4AaiabgUcaRiaaigdaaaGcqaaaaaaaaaWdbiabgEna0kabfo5ahn aaBaaaleaacaWGPbaabeaaaaa@4275@ with Γ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeu4KdC0aaSbaaSqaaiaadMgaaeqaaaaa@38E8@ defined in section 3.2), Uh = (U, V)T and n ^ Σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWHUb GbaKaajuaGdaWgaaqcbasaaKqzadGaeu4Odmfaleqaaaaa@3B20@ is the unitary vector perpendicular to Σ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqqHJo WujuaGdaWgaaqccasaaKqzadGaamyAaaadbeaaaaa@3B09@ , positive outward. Eq. (41.a) can be written as

Γ i ( ( U h h ˜ i k + 1 ) n ^ Σ ) d l + A i ( w s , i w b , i ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aa8qeaO qaaKqbaoaabmaakeaajuaGdaqadaGcbaqcLbsacaWHvbqcfa4aaSba aKaaGeaajugWaiacOHZGObaakeqaaKqzGeGaaGjcVlqadIgagaacaS Waa0baaKaaGeaajugWaiaadMgaaKaaGeaajugWaiaadUgacqGHRaWk caaIXaaaaaGccaGLOaGaayzkaaqcLbsacqGHflY1ceWHUbGbaKaaju aGdaWgaaqcaasaaKqzadGaeu4OdmfakeqaaaGaayjkaiaawMcaaKqz GeGaaGjcVlaayIW7caWGKbGaamiBaaGcbaqcLbsacqqHtoWrjuaGda WgaaqcaasaaKqzadGaiaiGC=pC=pyAaaGcbeaaaeqajugibiabgUIi YdGaey4kaSIaamyqaKqbaoaaBaaajaaibaqcLbmacGaGenyAaaGcbe aajuaGdaqadaGcbaqcLbsacaWG3bqcfa4aaSbaaKaaGeaajugWaiaa dohacaGGSaGaamyAaaGcbeaajugibiabgkHiTiaadEhajuaGdaWgaa qcaasaaKqzadGaamOyaiaacYcacaWGPbaakeqaaaGaayjkaiaawMca aKqzGeGaeyypa0JaaGimaaaa@794D@ (41.b).

Let l = 1 , N l i F ˜ l , i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqqFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaabeaO qaaKqzGeGabmOrayaaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGG SaGaamyAaaWcbeaaaKqaGeaajugWaiaadYgacqGH9aqpcaaIXaGaai ilaiaad6eacaWGSbWcdaWgaaqccasaaKqzadGaamyAaaqccasabaaa leqajugibiabggHiLdaaaa@47D7@ be the first term on the left hand side (l.h.s.) of Eq. (41.b), with F ˜ l , i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGgb GbaGaajuaGdaWgaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaaleqa aaaa@3BFE@ the flux across the l-th side of Γ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeu4KdC0aaSbaaSqaaiaadMgaaeqaaaaa@38E8@ , computed by Eq. (18) at the end of the hydrostatic DC step (with q ˜ s k + 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGXb GbaGaalmaaDaaajeaibaqcLbmacaWGZbaajeaibaqcLbmacaWGRbGa ey4kaSIaaGymaaaaaaa@3DB5@ , s = x, y).

According to Eq. (7), the z momentum equation (Eq. (6.c)) becomes

( W h ) t = 1 2 ( ( h w s ) t + ( h w b ) t ) = q b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaa GcbaqcLbsacqGHciITjuaGdaqadaGcbaqcLbsacaWGxbGaamiAaaGc caGLOaGaayzkaaaabaqcLbsacqGHciITcaWG0baaaiabg2da9Kqbao aalaaakeaajugibiaaigdaaOqaaKqzGeGaaGOmaaaajuaGdaqadaGc baqcfa4aaSaaaOqaaKqzGeGaeyOaIyBcfa4aaeWaaOqaaKqzGeGaam iAaiaadEhajuaGdaWgaaqcbauaaKqzGdGaam4CaaWcbeaaaOGaayjk aiaawMcaaaqaaKqzGeGaeyOaIyRaamiDaaaacqGHRaWkjuaGdaWcaa GcbaqcLbsacqGHciITjuaGdaqadaGcbaqcLbsacaWGObGaam4DaKqb aoaaBaaajeaibaqcLbmacaWGIbaaleqaaaGccaGLOaGaayzkaaaaba qcLbsacqGHciITcaWG0baaaaGccaGLOaGaayzkaaqcLbsacqGH9aqp caWGXbqcfa4aaSbaaKqaGeaajugWaiaadkgaaSqabaaaaa@67E8@ (42).

After discretization in time, we approximate Eq. (42) as

h ˜ k + 1 2 Δ t ( w s k + 1 w s k + w b k + 1 w b k ) = q b k + 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajuaGdaWcaa GcbaqcLbsaceWGObGbaGaajuaGdaahaaWcbeqcbasaaKqzadGaam4A aiabgUcaRiaaigdaaaaakeaajugibiaaikdacqqHuoarcaWG0baaaK qbaoaabmaakeaajugibiaadEhalmaaDaaajeaibaqcLbmacaWGZbaa jeaibaqcLbmacaWGRbGaey4kaSIaaGymaaaajugibiaayIW7cqGHsi slcaaMi8Uaam4DaSWaa0baaKqaGeaajugWaiaadohaaKqaGeaajugW aiaadUgaaaqcLbsacaaMi8Uaey4kaSIaaGjcVlaadEhalmaaDaaaje aibaqcLbmacaWGIbaajeaibaqcLbmacaWGRbGaey4kaSIaaGymaaaa jugibiaayIW7cqGHsislcaaMi8Uaam4DaSWaa0baaKqaGeaajugWai aadkgaaKqaGeaajugWaiaadUgaaaaakiaawIcacaGLPaaajugibiab g2da9iaadghalmaaDaaajeaibaqcLbmacaWGIbaajeaibaqcLbmaca WGRbGaey4kaSIaaGymaaaaaaa@7465@ (43),

the bottom vertical velocity as (see the first of Eqs. (3))

w b k + 1 w ˜ b k + 1 = ( ( U ˜ h ˜ ) k + 1 h ˜ k + 1 ) z b x + ( ( V ˜ h ˜ ) k + 1 h ˜ k + 1 ) z b y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWG3b WcdaqhaaqcKfaG=haajugWaiaadkgaaKazba4=baqcLbmacaWGRbGa ey4kaSIaaGymaaaajugibiabloKi7iqadEhagaacaSWaa0baaKazba 4=baqcLbmacaWGIbaajqwaa+FaaKqzadGaam4AaiabgUcaRiaaigda aaqcLbsacqGH9aqpjuaGdaqadaGcbaqcfa4aaSaaaOqaaKqbaoaabm aakeaajugibiqadwfagaacaiqadIgagaacaaGccaGLOaGaayzkaaqc fa4aaWbaaSqabKqaGeaajugWaiaadUgacqGHRaWkcaaIXaaaaaGcba qcLbsaceWGObGbaGaajuaGdaahaaWcbeqcbasaaKqzadGaam4Aaiab gUcaRiaaigdaaaaaaaGccaGLOaGaayzkaaqcfa4aaSaaaOqaaKqzGe GaeyOaIyRaamOEaKqbaoaaBaaajeaibaqcLbmacaWGIbaaleqaaaGc baqcLbsacqGHciITcaWG4baaaiabgUcaRKqbaoaabmaakeaajuaGda WcaaGcbaqcfa4aaeWaaOqaaKqzGeGabmOvayaaiaGabmiAayaaiaaa kiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaKqzadGaam4AaiabgU caRiaaigdaaaaakeaajugibiqadIgagaacaKqbaoaaCaaaleqajeai baqcLbmacaWGRbGaey4kaSIaaGymaaaaaaaakiaawIcacaGLPaaaju aGdaWcaaGcbaqcLbsacqGHciITcaWG6bqcfa4aaSbaaKqaGeaajugW aiaadkgaaSqabaaakeaajugibiabgkGi2kaadMhaaaaaaa@8821@ (44),

and the vertical velocity at the free surface is computed by Eq. (43), as

w s k + 1 = w b k w ˜ b k + 1 + w s k + 2 Δ t q b k + 1 h ˜ h + 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWG3b WcdaqhaaqcbasaaKqzadGaam4CaaqcbasaaKqzadGaam4AaiabgUca RiaaigdaaaqcLbsacqGH9aqpcaWG3bWcdaqhaaqcbasaaKqzadGaam OyaaqcbasaaKqzadGaam4AaaaajugibiabgkHiTiqadEhagaacaSWa a0baaKqaGeaajugWaiaadkgaaKqaGeaajugWaiaadUgacqGHRaWkca aIXaaaaKqzGeGaey4kaSIaam4DaSWaa0baaKqaGeaajugWaiaadoha aKqaGeaajugWaiaadUgaaaqcLbsacqGHRaWkjuaGdaWcaaGcbaqcLb sacaaMc8UaaGOmaiabfs5aejaadshacaaMc8UaamyCaSWaa0baaKqa GeaajugWaiaadkgaaKqaGeaajugWaiaadUgacqGHRaWkcaaIXaaaaa GcbaqcLbsaceWGObGbaGaajuaGdaahaaWcbeqcbasaaKqzadGaamiA aiabgUcaRiaaigdaaaaaaaaa@6B83@ (45).

The horizontal, dynamic components of the momentum equations are

q s t + 1 2 ( q b h ) s + q b z b s = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyRaamyCaKqbaoaaBaaajeaibaqcLbmacaWGZbaa leqaaaGcbaqcLbsacqGHciITcaWG0baaaiabgUcaRKqbaoaalaaake aajugibiaaigdaaOqaaKqzGeGaaGOmaaaajuaGdaWcaaGcbaqcLbsa cqGHciITjuaGdaqadaGcbaqcLbsacaWGXbqcfa4aaSbaaKqaGeaaju gWaiaadkgaaSqabaqcLbsacaWGObaakiaawIcacaGLPaaaaeaajugi biabgkGi2kaadohaaaGaey4kaSIaamyCaKqbaoaaBaaajeaibaqcLb macaWGIbaaleqaaKqbaoaalaaakeaajugibiabgkGi2kaadQhajuaG daWgaaqcbasaaKqzadGaamOyaaWcbeaaaOqaaKqzGeGaeyOaIyRaam 4CaaaacqGH9aqpcaaIWaaaaa@617F@ s = x, y (46).

After discretization in time, Eqs. (46) become

q s k + 1 = q ˜ s k + 1 Δ t 2 ( h ˜ k + 1 q b k + 1 s + q b k + 1 ( H ˜ k + 1 s + z b s ) ) = 0        s = { x y       q s k + 1 = { ( U h ) k + 1 ( V h ) k + 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGXb WcdaqhaaqcbasaaKqzadGaam4CaaqcbasaaKqzadGaam4AaiabgUca RiaaigdaaaqcLbsacaaMi8Uaeyypa0JaaGjcVlqadghagaacaSWaa0 baaKqaGeaajugWaiaadohaaKqaGeaajugWaiaadUgacqGHRaWkcaaI XaaaaKqzGeGaeyOeI0scfa4aaSaaaOqaaKqzGeGaeuiLdqKaamiDaa GcbaqcLbsacaaIYaaaaKqbaoaabmaakeaajugibiqadIgagaacaSWa aWbaaKqaGeqabaqcLbmacaWGRbGaey4kaSIaaGymaaaajuaGdaWcaa GcbaqcLbsacqGHciITcaWGXbWcdaqhaaqcbasaaKqzadGaamOyaaqc basaaKqzadGaam4AaiabgUcaRiaaigdaaaaakeaajugibiabgkGi2k aadohaaaGaey4kaSIaamyCaSWaa0baaKqaGeaajugWaiaadkgaaKqa GeaajugWaiaadUgacqGHRaWkcaaIXaaaaKqbaoaabmaakeaajuaGda WcaaGcbaqcLbsacqGHciITceWGibGbaGaajuaGdaahaaWcbeqcbasa aKqzadGaam4AaiabgUcaRiaaigdaaaaakeaajugibiabgkGi2kaado haaaGaey4kaSscfa4aaSaaaOqaaKqzGeGaeyOaIyRaamOEaKqbaoaa BaaajeaibaqcLbmacaWGIbaaleqaaaGcbaqcLbsacqGHciITcaWGZb aaaaGccaGLOaGaayzkaaaacaGLOaGaayzkaaqcLbsacqGH9aqpcaaI WaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaam4Caiabg2 da9KqbaoaaceaakeaajugibuaabeqaceaaaOqaaKqzGeGaamiEaaGc baqcLbsacaWG5baaaaGccaGL7baajugibiaabccacaqGGaGaeyO0H4 TaaeiiaiaabccacaWGXbWcdaqhaaqcbasaaKqzadGaam4Caaqcbasa aKqzadGaam4AaiabgUcaRiaaigdaaaqcLbsacqGH9aqpjuaGdaGaba GcbaqcLbsafaqabeGabaaakeaajuaGdaqadaGcbaqcLbsacaWGvbGa amiAaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabKqaGeaajugWaiaadU gacqGHRaWkcaaIXaaaaaGcbaqcfa4aaeWaaOqaaKqzGeGaamOvaiaa dIgaaOGaayjkaiaawMcaaKqbaoaaCaaaleqajeaibaqcLbmacaWGRb Gaey4kaSIaaGymaaaaaaaakiaawUhaaaaa@B583@ (47).

Symbol ( MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeyyXICnaaa@383F@ )k+1 marks the values of the variables at the end of the non-hydrostatic step.

Substituting Eqs. (44)-(45) and Eqs. (47) in Eq. (41,b), using the previous symbols, we get the following equation in the qb unknown

D i , 1 l = 1 , N l i q b k + 1 n i , i p d l , i Δ t 2 q b , i k + 1 l = 1 , N l i ( H ˜ k + 1 n i , i p + z b n i , i p ) d l , i + 2 Δ t h ˜ i k + 1 q b , i k + 1 A i = l = 1 , N l i F ˜ l , i A i ( w s , i k + w b , i k 2 w ˜ b , i k + 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqqFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGceaGabeaajugibi abgkHiTiqadseagaafaKqbaoaaBaaajeaibaqcLbmacaWGPbGaaiil aiaaigdaaSqabaqcfa4aaabuaOqaaKqbaoaalaaakeaajugibiabgk Gi2kaadghalmaaDaaajeaibaqcLbmacaWGIbaajeaibaqcLbmacaWG RbGaey4kaSIaaGymaaaaaOqaaKqzGeGaeyOaIyRaamOBaKqbaoaaBa aajeaibaqcLbmacaWGPbGaaiilaiaadMgacaWGWbaaleqaaaaajugi biaadsgajuaGdaWgaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaale qaaaqcbasaaKqzadGaamiBaiabg2da9iaaigdacaGGSaGaamOtaiaa dYgalmaaBaaajiaibaqcLbmacaWGPbaajiaibeaaaSqabKqzGeGaey yeIuoacqGHsisljuaGdaWcaaGcbaqcLbsacqqHuoarcaWG0baakeaa jugibiaaikdaaaGaaGPaVlaadghalmaaDaaajeaibaqcLbmacaWGIb GaaiilaiaadMgaaKqaGeaajugWaiaadUgacqGHRaWkcaaIXaaaaKqz GeGaaGjcVNqbaoaaqafakeaajuaGdaqadaGcbaqcfa4aaSaaaOqaaK qzGeGaeyOaIyRabmisayaaiaqcfa4aaWbaaSqabKqaGeaajugWaiaa dUgacqGHRaWkcaaIXaaaaaGcbaqcLbsacqGHciITcaWGUbqcfa4aaS baaKqaGeaajugWaiaadMgacaGGSaGaamyAaiaadchaaSqabaaaaKqz GeGaey4kaSscfa4aaSaaaOqaaKqzGeGaeyOaIyRaamOEaKqbaoaaBa aajeaibaqcLbmacaWGIbaaleqaaaGcbaqcLbsacqGHciITcaWGUbqc fa4aaSbaaKqaGeaajugWaiaadMgacaGGSaGaamyAaiaadchaaSqaba aaaaGccaGLOaGaayzkaaqcLbsacaWGKbqcfa4aaSbaaKqaGeaajugW aiaadYgacaGGSaGaamyAaaWcbeaaaKqaGeaajugWaiaadYgacqGH9a qpcaaIXaGaaiilaiaad6eacaWGSbWcdaWgaaqccasaaKqzadGaamyA aaqccasabaaaleqajugibiabggHiLdGaey4kaSscfa4aaSaaaOqaaK qzGeGaaGOmaiabfs5aejaadshaaOqaaKqzGeGabmiAayaaiaqcfa4a a0baaSqaaKqzGeGaamyAaaqcbasaaKqzadGaam4AaiabgUcaRiaaig daaaaaaKqzGeGaamyCaSWaa0baaKqaGeaajugWaiaadkgacaGGSaGa amyAaaqcbasaaKqzadGaam4AaiabgUcaRiaaigdaaaqcLbsacaWGbb qcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaqcLbsacqGH9aqpaOqa aKqzGeGaeyOeI0scfa4aaabuaOqaaKqzGeGabmOrayaaiaqcfa4aaS baaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaaaKqaGeaajugW aiaadYgacqGH9aqpcaaIXaGaaiilaiaad6eacaWGSbWcdaWgaaqcca saaKqzadGaamyAaaqccasabaaaleqajugibiabggHiLdGaeyOeI0Ia amyqaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqbaoaabmaake aajugibiaadEhalmaaDaaajeaibaqcLbmacaWGZbGaaiilaiaadMga aKqaGeaajugWaiaadUgaaaqcLbsacqGHRaWkcaWG3bWcdaqhaaqcba saaKqzadGaamOyaiaacYcacaWGPbaajeaibaqcLbmacaWGRbaaaKqz GeGaeyOeI0IaaGOmaiqadEhagaacaSWaa0baaKqaGeaajugWaiaadk gacaGGSaGaamyAaaqcbasaaKqzadGaam4AaiabgUcaRiaaigdaaaaa kiaawIcacaGLPaaaaaaa@FC7A@ i = 1, ..., N (48),

where D i , 1 = Δ t 2 h ˜ i k + 1   if     ϕ i k ϕ i p k             D i , 1 = Δ t 2 h ˜ i p k + 1    if     ϕ i p k > ϕ i k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGeb GbaqbajuaGdaWgaaqcbasaaKqzadGaamyAaiaacYcacaaIXaaaleqa aKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeuiLdqKaamiDaaGcba qcLbsacaaIYaaaaiqadIgagaacaSWaa0baaKqaGeaajugWaiaadMga aKqaGeaajugWaiaadUgacqGHRaWkcaaIXaaaaKqzGeGaaeiiaiaabc cacaqGPbGaaeOzaiaabccacaqGGaGaaeiiaiaabccacqaHvpGzlmaa DaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGRbaaaKqzGeGaey yzImRaeqy1dy2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaKqaGeaa jugWaiaadUgaaaGaaeiiaiaabccajugibiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiqadseagaafaKqb aoaaBaaajeaibaqcLbmacaWGPbGaaiilaiaaigdaaSqabaqcLbsacq GH9aqpjuaGdaWcaaGcbaqcLbsacqqHuoarcaWG0baakeaajugibiaa ikdaaaGabmiAayaaiaWcdaqhaaqcbasaaKqzadGaamyAaiaadchaaK qaGeaajugWaiaadUgacqGHRaWkcaaIXaaaaKqzGeGaaeiiaiaabcca caqGGaGaaeyAaiaabAgacaqGGaGaaeiiaiaabccacaqGGaGaeqy1dy 2cdaqhaaqcbasaaKqzadGaamyAaiaadchaaKqaGeaajugWaiaadUga aaqcLbsacqGH+aGpcqaHvpGzlmaaDaaajeaibaqcLbmacaWGPbaaje aibaqcLbmacaWGRbaaaaaa@923F@ (49),

and q b / n i , i p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSGbaO qaaKqzGeGaeyOaIyRaamyCaKqbaoaaBaaajeaibaqcLbmacaWGIbaa leqaaaGcbaqcLbsacqGHciITcaWGUbqcfa4aaSbaaKqaGeaajugWai aadMgacaGGSaGaamyAaiaadchaaSqabaaaaaaa@450B@ is the qb gradient along the n ^ i , i p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWHUb GbaKaajuaGdaWgaaqcbasaaKqzadGaamyAaiaacYcacaWGPbGaamiC aaWcbeaaaaa@3D1D@ direction. We set

F l , i = D i , 1 q b k + 1 n i , i p d l , i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aauaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa jugibiabg2da9iabgkHiTiqadseagaafaKqbaoaaBaaajeaibaqcLb macaWGPbGaaiilaiaaigdaaSqabaGcdaWcaaqcfayaaiabgkGi2kaa dghadaqhaaqcfasaaiaadkgaaeaacaWGRbGaey4kaSIaaGymaaaaaK qbagaacqGHciITcaWGUbWaaSbaaKqbGeaacaWGPbGaaiilaiaadMga caWGWbaajuaGbeaaaaqcLbsacaWGKbqcfa4aaSbaaKqaGeaajugWai aadYgacaGGSaGaamyAaaWcbeaaaaa@5711@ (50),

and it can be considered a (corrective) flux due to the difference of qb in cell i and in the neighboring ip one (its units are cubic meters per seconds, i.e., the ones of a volumetric flux).

As for the hydrostatic DC step, we adopt a linear P1 spatial approximation of the qb unknowns according to the nodal values, and the computational cell is the same as in the hydrostatic problem.

Similar to Eq. (37), we rewrite Eq. (50) as

F l , i = D i , 1 ( q b , i p k + 1 q b , i k + 1 ) | r i , i p | d l , i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aauaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa jugibiabg2da9iabgkHiTiqadseagaafaKqbaoaaBaaajeaibaqcLb macaWGPbGaaiilaiaaigdaaSqabaqcfa4aaeWaaOqaaKqzGeGaamyC aSWaa0baaKqaGeaajugWaiaadkgacaGGSaGaamyAaiaadchaaKqaGe aajugWaiaadUgacqGHRaWkcaaIXaaaaKqzGeGaeyOeI0IaamyCaSWa a0baaKqaGeaajugWaiaadkgacaGGSaGaamyAaaqcbasaaKqzadGaam 4AaiabgUcaRiaaigdaaaaakiaawIcacaGLPaaadaWccaqaaaqaaaaa jugibiaaysW7juaGdaabdaGcbaqcLbsacaWHYbqcfa4aaSbaaKqaGe aajugWaiaadMgacaGGSaGaamyAaiaadchaaSqabaaakiaawEa7caGL iWoajugibiaadsgajuaGdaWgaaqcbasaaKqzadGaamiBaiaacYcaca WGPbaaleqaaaaa@6D08@ (51).

As in the DC problem, if the GD property is satisfied for side ri,ip, dl,i is positive (see Eq. (10) and Appendix B), and the sign of F l , i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aauaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3B1A@ only depends on the difference of qb in the two cells i and ip, while the opposite occurs if the GD property is not satisfied.

Eqs. (48) form a linear system of order N in the qb,i (i = 1, …, N) unknowns. The diagonal and off-diagonal entries of the matrix are, respectively

s i , i = 2 Δ t A i h ˜ i k + 1 + l = 1 , N l i D i , 1 | r i , i p | d l , i l = 1 , N l i Δ t 2 ( n H ˜ k + 1 + n z b ) d l , i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGaam4CaK qbaoaaBaaajeaibaqcLbmacaWGPbGaaiilaiaadMgaaSqabaqcLbsa cqGH9aqpkmaalaaajuaGbaGaaGOmaiabfs5aejaadshacaaMc8Uaam yqamaaBaaajuaibaGaamyAaaqcfayabaaabaGabmiAayaaiaWaa0ba aKqbGeaacaWGPbaabaGaam4AaiabgUcaRiaaigdaaaaaaKqzGeGaey 4kaSscfa4aaabuaOqaamaalaaajuaGbaGabmirayaauaWaaSbaaKqb GeaacaWGPbGaaiilaiaaigdaaKqbagqaaaqaamaaemaabaGaaCOCam aaBaaajuaibaGaamyAaiaacYcacaWGPbGaamiCaaqcfayabaaacaGL hWUaayjcSdaaaKqzGeGaamizaKqbaoaaBaaajeaibaqcLbmacaWGSb GaaiilaiaadMgaaSqabaqcLbsacaaMc8oajeaibaqcLbmacaWGSbGa eyypa0JaaGymaiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugWai aadMgaaKGaGeqaaaWcbeqcLbsacqGHris5aiabgkHiTKqbaoaaqafa keaadaWcaaqcfayaaiabfs5aejaadshaaeaacaaIYaaaamaabmaake aajugibiabgEGirNqbaoaaBaaajeaibaqcLbmacaWGUbaaleqaaKqz GeGabmisayaaiaqcfa4aaWbaaSqabKqaGeaajugWaiaadUgacqGHRa WkcaaIXaaaaKqzGeGaey4kaSIaey4bIeDcfa4aaSbaaKqaGeaajugW aiac0b4GUbaaleqaaKqzGeGaamOEaKqbaoaaBaaajeaibaqcLbmaca WGIbaaleqaaaGccaGLOaGaayzkaaqcLbsacaaMi8UaaGPaVlaadsga juaGdaWgaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaaleqaaaqcba saaKqzadGaamiBaiabg2da9iaaigdacaGGSaGaamOtaiaadYgalmaa BaaajiaibaqcLbmacaWGPbaajiaibeaaaSqabKqzGeGaeyyeIuoaaa a@9F08@ s i , i p = D i , 1 | r i , i p | d l , i δ i , i p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaKqzGeGaam4CaK qbaoaaBaaajeaibaqcLbmacaWGPbGaaiilaiaadMgacaWGWbaaleqa aKqzGeGaeyypa0JaeyOeI0IcdaWcaaqcfayaaiqadseagaafamaaBa aajuaibaGaamyAaiaacYcacaaIXaaajuaGbeaaaeaadaabdaqaaiaa hkhadaWgaaqcfasaaiaadMgacaGGSaGaamyAaiaadchaaKqbagqaaa Gaay5bSlaawIa7aaaajugibiaadsgajuaGdaWgaaqcbasaaKqzadGa amiBaiaacYcacaWGPbaaleqaaKqzGeGaeqiTdqwcfa4aaSbaaKqaGe aajugWaiaadMgacaGGSaGaamyAaiaadchaaSqabaaaaa@59BB@ i, ip = 1, ..., N (52),

where n H ˜ k + 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiabgE GirNqbaoaaBaaajeaibaqcLbmacaWGUbaaleqaaKqzGeGabmisayaa iaqcfa4aaWbaaSqabKqaGeaajugWaiaadUgacqGHRaWkcaaIXaaaaa aa@4182@ and n z b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiabgE GirNqbaoaaBaaajeaibaqcLbmacaWGUbaaleqaaKqzGeGaamOEaKqb aoaaBaaajeaibaqcLbmacaWGIbaaleqaaaaa@3FFE@ are the spatial gradients of H and zb computed along the direction n ^ i , i p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWHUb GbaKaajuaGdaWgaaqcbasaaKqzadGaamyAaiaacYcacaWGPbGaamiC aaWcbeaaaaa@3D1D@ . The gradient of zb in cell i are computed as the gradient of H (see section 4.3). The other symbols are specified as above. The source term of system (48) is

b i = l = 1 , N l i F ˜ l , i A i ( w s , i k + w b , i k 2 w ˜ b , i k + 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGIb qcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaqcLbsacqGH9aqpcqGH sisljuaGdaaeqaGcbaqcLbsaceWGgbGbaGaajuaGdaWgaaqcbasaaK qzadGaamiBaiaacYcacaWGPbaaleqaaaqaaKqzGeGaamiBaiabg2da 9iaaigdacaGGSaGaamOtaiaadYgajuaGdaWgaaqccasaaKqzadGaam yAaaadbeaaaSqabKqzGeGaeyyeIuoacqGHsislcaWGbbqcfa4aaSba aKqaGeaajugWaiaadMgaaSqabaqcfa4aaeWaaOqaaKqzGeGaam4DaS Waa0baaKqaGeaajugWaiaadohacaGGSaGaamyAaaqcbasaaKqzadGa am4AaaaajugibiabgUcaRiaadEhalmaaDaaajeaibaqcLbmacaWGIb GaaiilaiaadMgaaKqaGeaajugWaiaadUgaaaqcLbsacqGHsislcaaI YaGabm4DayaaiaWcdaqhaaqcbasaaKqzadGaamOyaiaacYcacaWGPb aajeaibaqcLbmacaWGRbGaey4kaSIaaGymaaaaaOGaayjkaiaawMca aaaa@7167@ (53).
Once qb is computed in each cell i, the horizontal flow rate components are updated according to Eqs. (47) and the velocity of the free surface according to Eq. (45).

Similarly to the DC step (section 4.2), the matrix of system (48) is sparse, symmetric and, for a GD mesh, positive-definite, and system (48) is solved as system (35) (see section 4.2).

As for the corrective flux F ˜ l , i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgea0lrP0xe9Fve9 Fve9VXqaaeGabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3E66@ in the hydrostatic DC step, according to Eq. (51) and the definition of the coefficient D i , 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWGeb GbaqbajuaGdaWgaaqcbasaaKqzadGaamyAaiaacYcacaaIXaaaleqa aaaa@3BD2@ in (Eq. (49)), the flux F l , i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgea0lrP0xe9Fve9 Fve9VXqaaeGabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aauaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3E72@ leaving from cell i (ip) to ip (i) is equal to the flux entering cell ip (i) from cell i (ip). If F l , i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgea0lrP0xe9Fve9 Fve9VXqaaeGabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aauaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3E71@ is leaving from i to ip or vice versa depends on the difference ( q b , i k + 1 q b , i p k + 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FuK8c8fsY=Hrpy0dar=Jb9hs0dXdbPYxe9vr0=vr 0=LrWZqaaeGabiGaaiaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiaadghalmaaDaaajeaibaqcLbmacaWGIbGaaiilaiaadMga aKqaGeaajugWaiaadUgacqGHRaWkcaaIXaaaaKqzGeGaeyOeI0Iaam yCaSWaa0baaKqaGeaajugWaiaadkgacaGGSaGaamyAaiaadchaaKqa GeaajugWaiaadUgacqGHRaWkcaaIXaaaaaGccaGLOaGaayzkaaaaaa@4BF9@ (see Eq. (51)).

Boundary Conditions (B.C.)

As specified in section 3, at the free surface, the stress of the wind is neglected and the (atmospheric) pressure is set to zero. The friction at the bottom is approximated by the Manning equation, (see Eqs. (6.a)-(6.b)). We assign free slip condition at the impervious walls.

During the CP hydrostatic problem, we deal with the B.C. for those cells with incoming and leaving flux/momentum fluxes as in section 4.4 of [33] (for brevity we refer the reader to the mentioned paper). The external assigned incoming fluxes/momentum fluxes are considered in the CP step [31,33]. During the CC step, in the inlet/outlet cells, we set the incoming fluxes and momentum fluxes from the neighboring cells equal to the leaving ones. This means that, in the boundary cells, the solution of the convection prediction problem is not altered by the convection correction problem. The B.C. in the DC step is handled as in [33]. We set qb = 0 in the inlet and outlet cells and this represents a Dirichlet condition for the solution of system (48).

Model Properties

Preservation of the local and global mass balance

The hydrostatic and dynamic problems involved in the proposed procedure rely on the local mass balance. This is imposed, for each cell i, after the solution of the ODEs system of the governing equations in the CP and CC steps (see Eq. (24)). The systems solved in the hydrostatic DC and dynamic steps are derived by a local mass balance, obtained by embedding the momentum equations in the continuity equation (see Eqs. (33)-(34) for the DC step and Eqs. (41) and (48) for the dynamic problem). We show that the proposed solver also preserves the global mass balance in all its steps.

Summing all Eqs. (24) at the end of the CP problem, we get

i=1,N A i ( h i k+1/3 h i k ) =Δt i=1,N ( F ¯ i in F ¯ i out ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaabeaO qaaKqzGeGaamyqaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqb aoaabmaakeaajugibiaadIgalmaaDaaajeaibaqcLbmacaWGPbaaje aibaqcLbmacaWGRbGaey4kaSIaaGymaiaac+cacaaIZaaaaKqzGeGa eyOeI0IaamiAaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWai aadUgaaaaakiaawIcacaGLPaaaaKqaGeaajugWaiaadMgacqGH9aqp caaIXaGaaiilaiaad6eaaSqabKqzGeGaeyyeIuoacqGH9aqpcqqHuo arcaWG0bqcfa4aaabeaOqaaKqbaoaabmaakeaajuaGdaqdaaGcbaqc LbsacaWGgbaaaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWai aadMgacaWGUbaaaKqzGeGaeyOeI0scfa4aa0aaaOqaaKqzGeGaamOr aaaalmaaDaaajeaibaqcLbmacaWGPbaajeaibaqcLbmacaWGVbGaam yDaiaadshaaaaakiaawIcacaGLPaaaaKqaGeaajugWaiaadMgacqGH 9aqpcaaIXaGaaiilaiaad6eaaSqabKqzGeGaeyyeIuoaaaa@755A@  (54),

and, due to Eq. (25,b), for each internal couple of linked cells i and ip, the r.h.s. of Eq. (54) is equal to the difference between the boundary incoming and leaving fluxes. i.e.,

i=1,N A i ( h i k+1/3 h i k ) =Δt( F ¯ bou in F ¯ bou out ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaabeaO qaaKqzGeGaamyqaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqb aoaabmaakeaajugibiaadIgalmaaDaaajeaibaqcLbmacaWGPbaaje aibaqcLbmacaWGRbGaey4kaSIaaGymaiaac+cacaaIZaaaaKqzGeGa eyOeI0IaamiAaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWai aadUgaaaaakiaawIcacaGLPaaaaKqaGeaajugWaiaadMgacqGH9aqp caaIXaGaaiilaiaad6eaaSqabKqzGeGaeyyeIuoacqGH9aqpcqqHuo arcaWG0bqcfa4aaeWaaOqaaKqbaoaanaaakeaajugibiaadAeaaaWc daqhaaqcbasaaKqzadGaamOyaiaad+gacaWG1baajeaibaqcLbmaca WGPbGaamOBaaaajugibiabgkHiTKqbaoaanaaakeaajugibiaadAea aaWcdaqhaaqcbasaaKqzadGaamOyaiaad+gacaWG1baajeaibaqcLb macaWGVbGaamyDaiaadshaaaaakiaawIcacaGLPaaaaaa@70A1@  (55,a),

with F ¯ bou in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaamOraaaalmaaDaaajeaibaqcLbmacaWGIbGaam4Baiaa dwhaaKqaGeaajugWaiaadMgacaWGUbaaaaaa@3F8A@  and F ¯ bou out MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaamOraaaalmaaDaaajeaibaqcLbmacaWGIbGaam4Baiaa dwhaaKqaGeaajugWaiaad+gacaWG1bGaamiDaaaaaaa@4090@  the sum of the incoming (assigned) and leaving boundary fluxes. In a similar way, at the end of the CC step, due to the continuity of the mean-in-time flux between i and ip (similarly to Eq. (25.b)) and since we handle the assigned boundary incoming fluxes F ¯ bou in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaamOraaaalmaaDaaajeaibaqcLbmacaWGIbGaam4Baiaa dwhaaKqaGeaajugWaiaadMgacaWGUbaaaaaa@3F8A@  in the CP step (see section 6), after simple manipulations, we can easily show that

i=1,N A i ( h i k+2/3 h i k ) =Δt( F ¯ bou in F ¯ bou out ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaabeaO qaaKqzGeGaamyqaSWaaSbaaKqaGeaajugWaiaadMgaaKqaGeqaaKqb aoaabmaakeaajugibiaadIgalmaaDaaajeaibaqcLbmacaWGPbaaje aibaqcLbmacaWGRbGaey4kaSIaaGOmaiaac+cacaaIZaaaaKqzGeGa eyOeI0IaamiAaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWai aadUgaaaaakiaawIcacaGLPaaaaKqaGeaajugWaiaadMgacqGH9aqp caaIXaGaaiilaiaad6eaaSqabKqzGeGaeyyeIuoacqGH9aqpcqqHuo arcaWG0bqcfa4aaeWaaOqaaKqbaoaanaaakeaajugibiaadAeaaaWc daqhaaqcbasaaKqzadGaamOyaiaad+gacaWG1baajeaibaqcLbmaca WGPbGaamOBaaaajugibiabgkHiTKqbaoaanaaakeaajugibiaadAea aaWcdaqhaaqcbasaaKqzadGaamOyaiaad+gacaWG1baajeaibaqcLb macaWGVbGaamyDaiaadshaaaaakiaawIcacaGLPaaaaaa@703E@  (55.b),

where F ¯ bou out MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aa0aaaO qaaKqzGeGaamOraaaalmaaDaaajeaibaqcLbmacaWGIbGaam4Baiaa dwhaaKqaGeaajugWaiaad+gacaWG1bGaamiDaaaaaaa@4090@ is the sum of the leaving boundary flux computed at the end of the CC problem. In the hydrostatic DC problem, as specified at the end of section 4.2, the flux F ˜ l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgea0lrP0xe9Fve9 Fve9VXqaaeGabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3E66@  leaving from cell i (ip) to ip (i) is equal to the flux entering cell ip (i) from cell i (ip). This implies that the sum of all these fluxes is zero and, after simple manipulations, it can be show that

i=1,N A i ( h ˜ i k+1 h i k+2/3 ) =Δt F dir MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaabeaO qaaKqzGeGaamyqaKqbaoaaBaaajeaibaqcLbmacaWGPbaaleqaaKqb aoaabmaakeaajugibiqadIgagaacaSWaa0baaKqaGeaajugWaiaadM gaaKqaGeaajugWaiaadUgacqGHRaWkcaaIXaaaaKqzGeGaeyOeI0Ia amiAaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWaiaadUgacq GHRaWkcaaIYaGaai4laiaaiodaaaaakiaawIcacaGLPaaaaKqaGeaa jugWaiaadMgacqGH9aqpcaaIXaGaaiilaiaad6eaaSqabKqzGeGaey yeIuoacqGH9aqpcqqHuoarcaWG0bGaaGPaVlaadAeajuaGdaWgaaqc basaaKqzadGaamizaiaadMgacaWGYbaaleqaaaaa@61AB@  (55.c),

with Fdir the sum of the boundary fluxes computed at the cells with imposed Dirichlet conditions.

If we add Eq. (55,c) to Eq. (55,b), we obtain

i=1,N A i ( h ˜ i k+1 h i k ) =Δt( F ¯ bou in F ¯ bou out + F dir ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaabeaO qaaKqzGeGaamyqaSWaaSbaaKqaGeaajugWaiaadMgaaKqaGeqaaKqb aoaabmaakeaajugibiqadIgagaacaSWaa0baaKqaGeaajugWaiaadM gaaKqaGeaajugWaiaadUgacqGHRaWkcaaIXaaaaKqzGeGaeyOeI0Ia amiAaSWaa0baaKqaGeaajugWaiaadMgaaKqaGeaajugWaiaadUgaaa aakiaawIcacaGLPaaaaKqaGeaajugWaiaadMgacqGH9aqpcaaIXaGa aiilaiaad6eaaSqabKqzGeGaeyyeIuoacqGH9aqpcqqHuoarcaWG0b qcfa4aaeWaaOqaaKqbaoaanaaakeaajugibiaadAeaaaWcdaqhaaqc basaaKqzadGaamOyaiaad+gacaWG1baajeaibaqcLbmacaWGPbGaam OBaaaajugibiabgkHiTKqbaoaanaaakeaajugibiaadAeaaaWcdaqh aaqcbasaaKqzadGaamOyaiaad+gacaWG1baajeaibaqcLbmacaWGVb GaamyDaiaadshaaaqcLbsacqGHRaWkcaWGgbqcfa4aaSbaaKqaGeaa jugWaiaadsgacaWGPbGaamOCaaWcbeaaaOGaayjkaiaawMcaaaaa@75F8@ (55.d),

which can be assumed as a relationships representing the global mass balance at the end of the hydrostatic problem. Because for each couple of linked internal cells i and ip, the corrective dynamic flux F l,i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Bc9 vqaqpepm0xbbG8FaYxGuc9pgc9Grpy0=as0Fb9pgea0lrP0xe9Fve9 Fve9VXqaaeGabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOray aauaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGGSaGaamyAaaWcbeaa aaa@3E72@  leaving from i (ip) to ip (i) (in Eq. (51)) is equal to the flux entering ip (i) from i (ip) (see also the consideration at the end of section 5), the sum of all these internal fluxes is zero. On the other hand, the corrective boundary flux of the cells at the boundaries is zero, since we set qb = 0 as boundary Dirichlet assigned value (see section 6). This means that the non-hydrostatic step does not affect the global mass balances.

 Preservation of a general steady gtate (GSS) condition

We assume that the GSS condition is achieved if the two relationships in Eq. (56) hold in each cell:

h i t =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyRaamiAaKqbaoaaBaaajeaibaqcLbmacaWGPbaa leqaaaGcbaqcLbsacqGHciITcaWG0baaaiabg2da9iaaicdaaaa@40E2@  and q s,i t =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyRaamyCaKqbaoaaBaaajeaibaqcLbmacaWGZbGa aiilaiaadMgaaSqabaaakeaajugibiabgkGi2kaadshaaaGaeyypa0 JaaGimaaaa@4293@  s = x, y i = 1, ..., N (56).

The numerical procedure proposed for the solution of the (CP + CC) steps for a generic cell relies on (see sections 4 and 5): 1) the balance of the incoming and leaving fluxes with the time change of the storage term, for the continuity equation and, 2) the balance among the incoming and leaving momentum fluxes with the time change of the flow rates, with the resultant of the hydrostatic pressure distribution, along cell interfaces and bottom surfaces, and the resultant of the bottom friction term, for the momentum equations.

Given an initial condition in the domain, hi, and qs,i, i = 1, ..., N and s = x, y, the above considerations imply the two following conditions,
a) If the incoming fluxes in any cell i are equal to the leaving ones, then h does not change in time, i.e., from Eqs. (21) and (26),

if     1 Δt Δt l=1,N l i δ l,i F l,i r,out ( t )dt = l=1,N l i ( 1 δ l,i ) F ¯ l,i r,in      d h i dt A i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaqGPb GaaeOzaiaabccacaqGGaGaaeiiaiaabccajuaGdaWcaaGcbaqcLbsa caaIXaaakeaajugibiabfs5aejaadshaaaqcfa4aa8quaOqaaKqbao aaqafakeaajugibiabes7aKLqbaoaaBaaajeaibaqcLbmacaWGSbGa aiilaiaadMgaaSqabaqcLbsacaWGgbWcdaqhaaqcbasaaKqzadGaam iBaiaacYcacaWGPbaajeaibaqcLbmacaWGYbGaaiilaiaad+gacaWG 1bGaamiDaaaajuaGdaqadaGcbaqcLbsacaWG0baakiaawIcacaGLPa aajugibiaadsgacaWG0baajeaibaqcLbmacaWGSbGaeyypa0JaaGym aiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugWaiaadMgaaKGaGe qaaaWcbeqcLbsacqGHris5aaWcbaqcLbsacqqHuoarcaWG0baaleqa jugibiabgUIiYdGaeyypa0tcfa4aaabuaOqaaKqbaoaabmaakeaaju gibiaaigdacqGHsislcqaH0oazjuaGdaWgaaqcbasaaKqzadGaamiB aiaacYcacaWGPbaaleqaaaGccaGLOaGaayzkaaqcLbsaceWGgbGbae balmaaDaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaKqaGeaajugW aiaadkhacaGGSaGaamyAaiaad6gaaaaajeaibaqcLbmacaWGSbGaey ypa0JaaGymaiaacYcacaWGobGaamiBaSWaaSbaaKGaGeaajugWaiaa dMgaaKGaGeqaaaWcbeqcLbsacqGHris5aiaabccacaqGGaGaeyO0H4 TaaeiiaiaabccajuaGdaWcaaGcbaqcLbsacaWGKbGaamiAaKqbaoaa BaaajeaibaqcLbmacaWGPbaaleqaaaGcbaqcLbsacaWGKbGaamiDaa aacaWGbbqcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaqcLbsacqGH 9aqpcaaIWaaaaa@A0C4@  with symbol r = p, c (57),
b) If the incoming and leaving momentum fluxes in any cell i are balanced by the resultant of the hydrostatic pressure distribution along the cell interfaces and bottom surfaces, as well as by the resultant of the bottom stress term, then the specific flow rates in cell i do not change in time, i.e., from Eqs. (22) and (27),

if  ( 1 Δt Δt R i r,s ( t )dt ) δ r + 1 Δt Δt l=1,N l i δ l,i M l,i r,s,out ( t )dt = l=1,N l i ( 1 δ l,i ) M ¯ l,i r,s,in ( t )    d q s,i ( t ) dt A i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaqGPb GaaeOzaiaabccacaqGGaqcfa4aaeWaaOqaaKqbaoaalaaakeaajugi biaaigdaaOqaaKqzGeGaeuiLdqKaamiDaaaajuaGdaWdrbGcbaqcLb sacaWGsbWcdaqhaaqcbasaaKqzadGaamyAaaqcbasaaKqzadGaamOC aiaacYcacaWGZbaaaKqbaoaabmaakeaajugibiaadshaaOGaayjkai aawMcaaKqzGeGaamizaiaadshaaKqaGeaajugWaiabfs5aejaadsha aSqabKqzGeGaey4kIipaaOGaayjkaiaawMcaaKqzGeGaeqiTdqwcfa 4aaSbaaKqaGeaajugWaiaadkhaaSqabaqcLbsacqGHRaWkjuaGdaWc aaGcbaqcLbsacaaIXaaakeaajugibiabfs5aejaadshaaaqcfa4aa8 quaOqaaKqbaoaaqafakeaajugibiabes7aKLqbaoaaBaaajeaibaqc LbmacaWGSbGaaiilaiaadMgaaSqabaqcLbsacaWGnbWcdaqhaaqcba saaKqzadGaamiBaiaacYcacaWGPbaajeaibaqcLbmacaWGYbGaaiil aiaadohacaGGSaGaam4BaiaadwhacaWG0baaaKqbaoaabmaakeaaju gibiaadshaaOGaayjkaiaawMcaaKqzGeGaamizaiaadshaaKqaGeaa jugWaiaadYgacqGH9aqpcaaIXaGaaiilaiaad6eacaWGSbWcdaWgaa qccasaaKqzadGaamyAaaqccasabaaaleqajugibiabggHiLdaajeai baqcLbmacqqHuoarcaWG0baaleqajugibiabgUIiYdGaeyypa0tcfa 4aaabuaOqaaKqbaoaabmaakeaajugibiaaigdacqGHsislcqaH0oaz juaGdaWgaaqcbasaaKqzadGaamiBaiaacYcacaWGPbaaleqaaaGcca GLOaGaayzkaaqcLbsaceWGnbGbaebalmaaDaaajeaibaqcLbmacaWG SbGaaiilaiaadMgaaKqaGeaajugWaiaadkhacaGGSaGaam4CaiaacY cacaWGPbGaamOBaaaajuaGdaqadaGcbaqcLbsacaWG0baakiaawIca caGLPaaaaKqaGeaajugWaiaadYgacqGH9aqpcaaIXaGaaiilaiaad6 eacaWGSbWcdaWgaaqccasaaKqzadGaamyAaaqccasabaaaleqajugi biabggHiLdGaaeiiaiabgkDiElaabccajuaGdaWcaaGcbaqcLbsaca WGKbGaamyCaSWaaSbaaKqaGeaajugWaiaadohacaGGSaGaamyAaaqc basabaqcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaaaba qcLbsacaWGKbGaamiDaaaacaWGbbqcfa4aaSbaaKqaGeaajugWaiaa dMgaaSqabaqcLbsacqGH9aqpcaaIWaaaaa@CDCF@  (58),

with δ r = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH0oazk8aadaWgaaqcbasaaKqzadWdbiaadkhaaSWdaeqa aKqzGeWdbiabg2da9iaabccacaaIXaaaaa@3E50@ in the CP step, δ r = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH0oazk8aadaWgaaqcbasaaKqzadWdbiaadkhaaSWdaeqa aKqzGeWdbiabg2da9iaabccacaaIWaaaaa@3E4F@  in the CC step. Apex r in Eqs. (57)-(58) is equal to "p" (in the convective prediction step) or "c" (in the convective correction step) and in this second case the source term Ri is equal to zero (see section 4.1).

Moreover, due to the fixed bed assumption, if the water depths in all the i cells do not change in time, neither the water levels nor their spatial gradients will do. We will show that, if conditions a) and b) of the (CP + CC) steps hold, the hydrostatic DC step and the dynamic problem do not modify this steady state condition for each cell. If the flow rate components do not change in time during the (CP + CC) steps (condition b) holds), the second sum on the r.h.s. of Eq. (33) in the DC step is zero, and, after simple manipulations, the same equation can be written as:
A i η i t d A i l=1,N l i D ˜ i ( ηϑ ) n l,i d l,i =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqVepy0de9LqFHe9Lq pepeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaajuaGdaWdrb Gcbaqcfa4aaSaaaOqaaKqzGeGaeyOaIyRaeq4TdGwcfa4aaSbaaKqa GeaajugWaiaadMgaaSqabaaakeaajugibiabgkGi2kaadshaaaGaam izaiaadgeajuaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaaaKqaGeaa jugWaiaadgealmaaBaaajiaibaqcLbmacaWGPbaajiaibeaaaSqabK qzGeGaey4kIipacqGHsisljuaGdaaeqbGcbaqcLbsaceWGebGbaGaa juaGdaWgaaqcbasaaKqzadGaamyAaaWcbeaajuaGdaWcaaGcbaqcLb sacqGHciITjuaGdaqadaGcbaqcLbsacqaH3oaAcqGHsislcqaHrpGs aOGaayjkaiaawMcaaaqaaKqzGeGaeyOaIyRaamOBaKqbaoaaBaaaje aibaqcLbmacaWGSbGaaiilaiaadMgaaSqabaaaaKqzGeGaamizaKqb aoaaBaaajeaibaqcLbmacaWGSbGaaiilaiaadMgaaSqabaaajeaiba qcLbmacaWGSbGaeyypa0JaaGymaiaacYcacaWGobGaamiBaSWaaSba aKGaGeaajugWaiaadMgaaKGaGeqaaaWcbeqcLbsacqGHris5aiabg2 da9iaaicdaaaa@7811@  (59),

with η= H ˜ k+1 H k +2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiabeE 7aOjabg2da9iqadIeagaacaKqbaoaaCaaaleqajeaibaqcLbmacaWG RbGaey4kaSIaaGymaaaajugibabaaaaaaaaapeGaeyOeI0IaamisaK qba+aadaahaaWcbeqaaKqzGeWdbiaadUgaaaqcfa4damaaCaaaleqa jeaibaqcLbmapeGaey4kaSIaaGOmaiaac+cacaaIZaaaaaaa@491C@  and θ= H k H k +2/3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH4oqCcqGH9aqpcaWGibGcpaWaaWbaaSqabKqaGeaajugW a8qacaWGRbaaaKqzGeGaeyOeI0IaamisaSWdamaaCaaajeaibeqaaK qzadWdbiaadUgaaaWcpaWaaWbaaKqaGeqabaqcLbmapeGaey4kaSIa aGOmaiaac+cacaaIZaaaaaaa@46AB@ . If h (H) in all the i cells does not change in time during the (CP + CC) steps (condition a) holds), θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH4oqCaaa@38AB@ is zero, and the solution of Eq. (59) is identically η=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiabeE 7aOjabg2da9iaaicdaaaa@3A58@ , i.e., zero water level correction. From Eq. (31) we get that the correction of the specific flow rate components after the DC step is zero. In brief, if conditions a) and b) hold, we get:
c) H ˜ k+1 = H k +2/3 =  H k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiqadI eagaacaKqbaoaaCaaaleqajeaibaqcLbmacaWGRbGaey4kaSIaaGym aaaajugibabaaaaaaaaapeGaeyypa0JaamisaSWdamaaCaaajeaibe qaaKqzadWdbiaadUgaaaWcpaWaaWbaaKqaGeqabaqcLbmapeGaey4k aSIaaGOmaiaac+cacaaIZaaaaKqzGeGaeyypa0JaaiiOaiaadIeaju aGpaWaaWbaaSqabKqaGeaajugWa8qacaWGRbaaaaaa@4CD8@  for each cell i,
d) q ˜ s k+1 = q s k+2/3 = q s k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiqadg hagaacaSWaa0baaKqaGeaajugWaiaadohaaKqaGeaajugWaiaadUga cqGHRaWkcaaIXaaaaKqzGeGaeyypa0JaamyCaSWaa0baaKqaGeaaju gWaiaadohaaKqaGeaajugWaiaadUgacqGHRaWkcaaIYaGaai4laiaa iodaaaqcLbsacqGH9aqpcaWGXbWcdaqhaaqcbasaaKqzadGaam4Caa qcbasaaKqzadGaam4Aaaaaaaa@5001@ , s = x, y for each cell i.
We prove that the correction due to the non-hydrostatic problem is zero as well. We focus our attention on the r.h.s. of Eq. (48). We find the two following conditions e) and f),
e) If conditions b) and d) hold, the sum of the horizontal fluxes is zero for any i cell, i.e., l=1,N l i F ˜ l,i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaabeaO qaaKqzGeGabmOrayaaiaqcfa4aaSbaaKqaGeaajugWaiaadYgacaGG SaGaamyAaaWcbeaaaKqaGeaajugWaiaadYgacqGH9aqpcaaIXaGaai ilaiaad6eacaWGSbWcdaWgaaqccasaaKqzadGaamyAaaqccasabaaa leqajugibiabggHiLdGaeyypa0JaaGimaaaa@49D7@ ,
f) If conditions a) to d) hold, q ˜ s k+1 / h ˜ k+1 = q s k / h k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqqFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcfa4aaSGbaO qaaKqzGeGabmyCayaaiaWcdaqhaaqcbasaaKqzadGaam4Caaqcbasa aKqzadGaam4AaiabgUcaRiaaigdaaaaakeaajugibiqadIgagaacaK qbaoaaCaaaleqajeaibaqcLbmacaWGRbGaey4kaSIaaGymaaaaaaqc LbsacqGH9aqpjuaGdaWcgaGcbaqcLbsacaWGXbWcdaqhaaqcbasaaK qzadGaam4CaaqcbasaaKqzadGaam4AaaaaaOqaaKqzGeGaamiAaKqb aoaaCaaaleqajeaibaqcLbmacaWGRbaaaaaaaaa@51A4@  (s = x, y).

From condition-(f) and Eqs. (3), we obtain that w ˜ s( b ) k+1 = w s( b ) k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWG3b GbaGaalmaaDaaajeaibaqcLbmacaWGZbWcdaqadaqcbasaaKqzadGa amOyaaqcbaIaayjkaiaawMcaaaqaaKqzadGaam4AaiabgUcaRiaaig daaaqcLbsacqGH9aqpcaWG3bWcdaqhaaqcbasaaKqzadGaam4CaSWa aeWaaKqaGeaajugWaiaadkgaaKqaGiaawIcacaGLPaaaaeaajugWai aadUgaaaaaaa@4C98@ . Integrating the divergence-free continuity equation (Eq. (1)) to a control volume corresponding to the generic computational cell (see section 5), From condition e) we get w ˜ s k+1 = w ˜ b k+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWG3b GbaGaalmaaDaaajeaibaqcLbmacaWGZbaajeaibaqcLbmacaWGRbGa ey4kaSIaaGymaaaajugibiabg2da9iqadEhagaacaSWaa0baaKqaGe aajugWaiaadkgaaKqaGeaajugWaiaadUgacqGHRaWkcaaIXaaaaaaa @468D@ .

According to these considerations, the r.h.s. of Eqs. (48) is zero and this leads to qb = 0. This implies that the dynamic correction of the flow rate components, with respect to the initial condition, is zero and it is consistent with the second condition in Eq. (54).

The "water at rest" condition [44] is a particular case of the general situation described as above, where the flow rate components qs,i (s = x, y) are zero for all the i cells. In this case, according to the argumentations as above, it easy to prove the C-property of the present model [44].

Handling of wetting/drying processes

In [32] the authors show 1) how negative water depths do not arise during the solution of the (CP + CC) steps and 2) how to handle small negative values of water depth calculated after the solution of the linear system of the DC step. If, after the solution of the hydrostatic step, we have, in cell i, h i 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqqFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aaSbaaKqaGeaajugWaiaadMgaaSqabaqcLbsacqGHKjYOcaaI Waaaaa@3D2F@ , we set qb = 0, as Dirichlet value in system (48).

 Study of the frequency dispersion

We compare the phase speed of the proposed model with the one given by the linear wave theory. The phase speed from the linear wave theory is [34]:

c 1 = gh ( tanh( k w h )/ k w h ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJb qcfa4aaSbaaKqaGeaajugWaiaaigdaaSqabaqcLbsacqGH9aqpjuaG daGcaaGcbaqcLbsacaWGNbGaamiAaaWcbeaajuaGdaGcaaGcbaqcfa 4aaeWaaOqaaKqbaoaalyaakeaajugibiGacshacaGGHbGaaiOBaiaa cIgajuaGdaqadaGcbaqcLbsacaWGRbqcfa4aaSbaaKqaGeaajugWai aadEhaaSqabaqcLbsacaWGObaakiaawIcacaGLPaaaaeaajugibiaa dUgajuaGdaWgaaqcbasaaKqzadGaam4DaaWcbeaajugibiaadIgaaa aakiaawIcacaGLPaaaaSqabaaaaa@54A9@  (60.a)

where kw is the wave number ( k w = 2π/ L w , L w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Aa8aadaWgaaWcbaWdbiaadEhaa8aabeaak8qacqGH9aqpcaqG GaGaaGOmaiabec8aWjaac+cacaWGmbWdamaaBaaaleaapeGaam4Daa WdaeqaaOWdbiaacYcacaWGmbWdamaaBaaaleaapeGaam4DaaWdaeqa aaaa@42B3@  is the wave length). The wave period Tw is equal to Lw/c1. We assume a standing regular wave with Lw = 20 m in a closed basin, 40 m long (= 2 Lw) and 1 m wide, with frictionless and horizontal bottom. The wave profile is:

η( x,0 )= a w cos( 2πx/ L w ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaH3o aAjuaGdaqadaGcbaqcLbsacaWG4bGaaiilaiaaicdaaOGaayjkaiaa wMcaaKqzGeGaeyypa0JaamyyaKqbaoaaBaaajeaibaqcLbmacaWG3b aaleqaaKqzGeGaci4yaiaac+gacaGGZbqcfa4aaeWaaOqaaKqbaoaa lyaakeaajugibiaaikdacqaHapaCcaWG4baakeaajugibiaadYeaju aGdaWgaaqcbasaaKqzadGaam4DaaWcbeaaaaaakiaawIcacaGLPaaa aaa@513D@  (60,b),

where η MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4TdGgaaa@3812@  is the distance between the still water level and the free surface and aw is the wave amplitude. We keep constant the value for aw, equal to 0.1 m and h ranges from 1 m to 20 m. This experiment is similar to the one performed in [11,16,18], where the authors assumed a channel length equal to Lw /2. We discretize the domain with equilateral triangles with side 0.1 m. The wave period is divided in 160 parts and the total simulation time is 30 Tw. We registered the computed η MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4TdGgaaa@3812@  elevation at two gauges, placed along the centerline of the channel, far Lw /2 and 3/2 Lw from the upstream end, so that the distance between the two gauges is equal to the wave length. We compute the numerical phase speed cn = Lw/Tw,n, where Tw,n is the time measured between two consecutive wave peaks of the same gauge, the first in the exact η( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH3oaAk8aadaqadaqaaKqzGeWdbiaadshaaOWdaiaawIca caGLPaaaaaa@3BF4@  profile, the second in the computed η( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaH3oaAk8aadaqadaqaaKqzGeWdbiaadshaaOWdaiaawIca caGLPaaaaaa@3BF4@  profile (Figure 3.a). In Figure 3.b we compare the numerical and the linear wave theory phase speeds versus kh, and we also plot the phase celerity analytically computed for the shallow waters limit (e.g., c = (gh)1/2. The numerical phase speed plotted in Figure 3.b is obtained as the mean value between the phases speed computed for the two gauges, which have very similar values. The numerical phase celerity begins to deviate from the linear theory at about kh = 2.74, i.e., for h/Lw = 0.436, corresponding to the "intermediate waters" [34]), similarly to the weakly dispersive BTMs [11,45]. Setting Tw,n as the time interval between two consecutive wave peaks computed at gauges G1 and G2, we obtained results very similar to the ones in Figure 3.b. A small increment of cn has been computed between kh = 4 - 5 (Figure 3.b). According to the numerical results, in this interval, the numerical solution progressively deviates from the linear theory, and a slight increment of the phase speed is balanced by a reduction of the computed wave peaks. For kh > 5, cn monotonically decreases.

Figure 3: a) Computation of the numerical phase celerity. b) Phase speed vs. water depth.

Model Applications

Test 1: Propagation of a solitary wave in a frictionless and horizontal channel

We deal with a solitary wave, travelling in a horizontal and frictionless channel, 850 m long and 10 m wide. The initial water level h0 is 10 m, the amplitude wave aw is 2 m and the wave is initially centered at x = 200 m. The initial conditions are given by the analytical solution [11,13,37].

η( x,t )= a w cosh 2 ( k w ( x x 0 ct ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiabeE 7aOLqbaoaabmaakeaajugibiaadIhacaGGSaGaamiDaaGccaGLOaGa ayzkaaqcLbsacqGH9aqpjuaGdaWcaaGcbaqcLbsacaWGHbqcfa4aaS baaKqaGeaajugWaiaadEhaaSqabaaakeaajugibiGacogacaGGVbGa ai4CaiaacIgajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajuaGda qadaGcbaqcLbsacaWGRbqcfa4aaSbaaKqaGeaajugWaiaadEhaaSqa baqcfa4aaeWaaOqaaKqzGeGaamiEaiabgkHiTiaadIhajuaGdaWgaa qcbasaaKqzadGaaGimaaWcbeaajugibiabgkHiTiaadogacaWG0baa kiaawIcacaGLPaaaaiaawIcacaGLPaaaaaaaaa@5E57@   U( x,t )= ηc η+ h 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiaadw fajuaGdaqadaGcbaqcLbsacaWG4bGaaiilaiaadshaaOGaayjkaiaa wMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeq4TdGMaam4yaa GcbaqcLbsacqaH3oaAcqGHRaWkcaWGObqcfa4aaSbaaKqaGeaajugW aiaaicdaaSqabaaaaaaa@4966@   w s ( x,t )= z| z= h 0 +η U x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiaadE hajuaGdaWgaaqcbasaaKqzadGaam4CaaWcbeaajuaGdaqadaGcbaqc LbsacaWG4bGaaiilaiaadshaaOGaayjkaiaawMcaaKqzGeGaeyypa0 JaeyOeI0scfa4aaqGaaOqaaKqzGeGaamOEaaGccaGLiWoajuaGdaWg aaWcbaqcLbsacaWG6bGaeyypa0JaamiAaKqbaoaaBaaajiaibaqcLb macaaIWaaameqaaKqzGeGaey4kaSIaeq4TdGgaleqaaKqbaoaalaaa keaajugibiabgkGi2kaadwfaaOqaaKqzGeGaeyOaIyRaamiEaaaaaa a@573D@  (61),

where η MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiabeE 7aObaa@3898@  is the elevation of the free surface measured from h0, c is the wave celerity c= g( h 0 + a w ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiaado gacqGH9aqpjuaGdaGcaaGcbaqcLbsacaWGNbqcfa4aaeWaaOqaaKqz GeGaamiAaKqbaoaaBaaajeaibaqcLbmacaaIWaaaleqaaKqzGeGaey 4kaSIaamyyaKqbaoaaBaaajeaibaqcLbmacaWG3baaleqaaaGccaGL OaGaayzkaaaaleqaaaaa@46E0@ , kw is the wave number k w = ( 3 a w / 4 h 0 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiaadU gajuaGdaWgaaqcbasaaKqzadGaam4DaaWcbeaajugibiabg2da9Kqb aoaakaaakeaajuaGdaqadaGcbaqcfa4aaSGbaOqaaKqzGeGaaG4mai aadggajuaGdaWgaaqcbasaaKqzadGaam4DaaWcbeaaaOqaaKqzGeGa aGinaiaadIgalmaaDaaajeaibaqcLbmacaaIWaaajeaibaqcLbmaca aIZaaaaaaaaOGaayjkaiaawMcaaaWcbeaaaaa@4BE3@  and the other symbols are specified as above. Left and right sides of the channel are open boundaries; the other two sides are impervious walls. The GD mesh used to discretize the channel has 160925 triangles and 84409 nodes. The mesh generator Netgen [46] has been used to create the meshes in all the applications presented in this paper. The size of the time step Δt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeyiLdqKaamiDaaaa@38C6@  is 0.025 s. The maximum value of the CFL number we computed is 1.925, where the CFL number for cell i is obtained as

CFL= ( ( Uh ) i 2 + ( Vh ) i 2 / h i + g h i )Δt/ A i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiaado eacaWGgbGaamitaiabg2da9KqbaoaalyaakeaajuaGdaqadaGcbaqc fa4aaSGbaOqaaKqbaoaakaaakeaajuaGdaqadaGcbaqcLbsacaWGvb GaamiAaaGccaGLOaGaayzkaaWcdaqhaaqcbasaaKqzadGaamyAaaqc basaaKqzadGaaGOmaaaajugibiabgUcaRKqbaoaabmaakeaajugibi aadAfacaWGObaakiaawIcacaGLPaaalmaaDaaajeaibaqcLbmacaWG PbaajeaibaqcLbmacaaIYaaaaaWcbeaaaOqaaKqzGeGaamiAaKqbao aaBaaajeaibaqcLbmacaWGPbaaleqaaaaajugibiabgUcaRKqbaoaa kaaakeaajugibiaadEgacaWGObqcfa4aaSbaaKqaGeaajugWaiaadM gaaSqabaaabeaaaOGaayjkaiaawMcaaKqzGeGaeuiLdqKaamiDaaGc baqcfa4aaOaaaOqaaKqzGeGaamyqaKqbaoaaBaaajeaibaqcLbmaca WGPbaaleqaaaqabaaaaaaa@660E@  (62).

In Figure 4 we show the initial condition for η MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4TdGgaaa@3812@ , U and ws and the solutions computed after 10, 20 and 50 s. The computed absolute values of the peaks slightly decrease at the very beginning, since the initial condition is given by an analytical solution and other authors observed similar trends [8,18]. The computed peak values of η MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4TdGgaaa@3812@  and U after 10 s are 1.926 m and 1.677 m/s, respectively. The maximum and minimum computed values of ws at the same time are 0.546 and -0.552 m/s, respectively. These values and shape of the computed profiles are almost saved at 50 s.

Figure 4: Test 1. H (top), U (middle) and ws (bottom). Initial solution and computed solutions after 10, 20 and 50 s.

Test 2: Moving shoreline inside a parabolic bowl. Investigation of the computational costs and convergence order

Sampson et al. [47] provided for this case the analytical solution for H, U and V (see Eqs. (39)-(40) in [48]) The domain is [8000]2 m2 with symmetric bottom with respect to the centre of the squared domain (see Eq. (38) in [48]). The GD mesh we use for spatial discretization has 272 elements and 149 nodes, and Δt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibiabfs 5aejaadshaaaa@394B@  is 40 s (we adopt the same mesh and time discretizations as in a previous paper [33], where we present a hydrostatic NLSEWs solver). The maximum CFL value obtained during the simulation is 2.82. We refine the computational mesh three times. Each refinement is performed by halving each sides of the previous coarser mesh (i.e., each triangle is divided in four equal ones). We limit the growth of the CFL number by halving the time step at each refinement. We compute the L2 norms of the relative errors of h, Uh and Vh with respect to the exact values (Table 1). The relative error for the mesh level l, errl and the rate of convergence rc, are computed as in Eqs. (70,a) and (70,b) in [33], respectively. The values of the rc are close to 2 (Table 1). The reason is that inside each triangle of the mesh, h (H), Uh and Vh change linearly, according to the three nodal values (2nd spatial approximation order). Refining the mesh, rc does not change significantly and this implies that the numerical model computes accurate results also if the domain in discretized using a coarse mesh instead of a very fine one.

Refinement
Level

NT

N

L2,h

rc,h

L2,Uh

rc,Uh

L2,Vh

rc,Vh

0

272

159

3.71E-02

5.16E+00

3.31E+00

1

1088

589

1.10E-02

1.75E+00

1.60E+00

1.69E+00

1.00E+00

1.73E+00

2

4352

2265

2.90E-03

1.92E+00

4.60E-01

1.80E+00

2.70E-01

1.89E+00

3

17408

8881

7.46E-04

1.96E+00

1.20E-01

1.94E+00

7.00E-02

1.95E+00

Table 1: Test 2. L2 norms of relative errors and rate of convergence rc.

We use this test to investigate the computational costs (CPU times). Table 2 shows the specific mean values of the CPU times, CPUs, required for cells ordering and for the solution of the (CP+CC) steps, DC step and non-hydrostatic problem. CPUs are computed by dividing the total CPU time required for the solution of the different steps of the algorithm, by the number of the cells and the time iterations. We use a single processor Intel CORE i7-4770 at 3.40 GHz. The (CP+CC) steps are the most demanding ones, and the small decrement of the CPUs can be motivated as in section 5.8 in [33]. The DC step, the non-hydrostatic problem and the cells ordering step require CPUs approximately one magnitude order less than the one required for (CP+CC) steps.

We also measure the mean CPU times per iteration, obtained by dividing the total CPU times of the different steps of the algorithm, by the number of the time iterations. We compute the growth rate β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqOSdigaaa@3807@  of the mean CPU time as in Eq. (71) in [33]. Results are shown in Figure 5 and the growth exponents β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqOSdigaaa@3807@  are close to 1. The value of the (CP + CC) steps, slightly less than 1, can be due to the small decrement of the computational effort observed with increasing the cells number, discussed in [32]. In the same figure we also show the total CPU times, obtained by adding the mean CPU times of the four algorithm steps, and the value of its β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqOSdigaaa@3807@  is very close to 1.

Refinement Level

NT

Cells Ordering

CP + CC

DC

Non-Hydrostatic

0

272

2.81E-07

1.19E-05

7.64E-07

5.36E-07

1

1088

3.15E-07

1.21E-05

8.63E-07

6.15E-07

2

4352

3.33E-07

1.18E-05

8.44E-07

6.72E-07

3

17408

3.40E-07

1.17E-05

9.05E-07

6.76E-07

Table 2: CPUs times (seconds).

Figure 5: Mean CPU times and exponents β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqOSdi gaaa@3876@

In Table D.1 (in Appendix D), we show the L2 norms of the relative errors of h, Uh and Vh, as well as the corresponding convergence orders, computed by the present solver without the non-hydrostatic solver. The norms are slightly greater than the ones obtained with the present non-hydrostatic solver, probably because of the lack of the correction of the flow field due to the components of the dynamic pressure. It is also important to compare the values in Table D.1 with the ones computed by the hydrostatic solver in [33] (Table 1 in [33].). In the solver of the referred paper, a cell-centered formulation is adopted, with a constant (1st order) spatial approximation order of the unknown variables h, Uh and Vh inside the computational cells (the triangles of the mesh). Changing from a node-centered to a cell-centered formulation, the number of the computational cells and unknowns increases. So we would expect a reduction of the errors in the solution of the cell-centered formulation. The reason of the smaller values of the errors of the present algorithm, compared to the ones in [33], could be due to the difference of the spatial approximation order of the unknown variables in the two adopted formulations.

Another benefit of the present solver with respect to the one in [33], is the abatement of the mean CPU times. In Table D.2 (in Appendix D), we list the CPUs of the solver in [33], a bit smaller than the ones in Table 1 of the present model. The reason could be the higher number of neighboring computational cells in the node-centered formulation, with respect to the three (or less, for the boundary triangles) in the cell-centered formulation. Due to the greater number of the computational cells, the total CPU times required by the cell-centered formulation (in Figure D.1, Appendix D), are greater than the ones of the present node-centered formulation (Figure 5).

Test 3: Propagation and breaking of a solitary wave over a sloping bottom

A solitary wave with ratio aw/h0 = 0.3, travels, breaks and runs up on a sloping plane beach. The measures provided in [49] constitute the reference solution for the one provided by the present model. The channel consists in a first horizontal stretch, followed by a beach with slope 1:19.85. The solitary wave in the numerical experiment is initially centered at half wavelength Lw from the beach toe [49],

L w =2/ k w arccosh( 20 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qcfa4aaSbaaKqaGeaajugWaiaadEhaaSqabaqcLbsacqGH9aqpjuaG daWcgaGcbaqcLbsacaaIYaaakeaajugibiaadUgajuaGdaWgaaqcba saaKqzadGaam4DaaWcbeaaaaqcLbsacaqGHbGaaeOCaiaabogacaqG JbGaae4BaiaabohacaqGObqcfa4aaeWaaOqaaKqbaoaakaaakeaaju gibiaaikdacaaIWaaaleqaaaGccaGLOaGaayzkaaaaaa@4E72@  (63),

For the numerical experiments, we use a 120 m long and 1.2 m wide channel, discretized with a GD mesh with 24234 triangles and 13282 nodes. The toe of the beach is 60 m far from the left channel end. We assign zero flux at the right side of the domain, and the boundary conditions at the other sides of the domain, as well as the initial conditions, are as in test 1. The size of the time step is 0.016 s and the maximum computed value of CFL is 1.17. In Figure 6 we compare the simulated and the measured free surface elevations. The length and time scales are normalized by the initial still water level h0 and (h0/g)1/2, respectively. When the wave propagates over the inclined beach, its front skews and, according to the experimental data, it breaks around t(g/h0)1/2 = 20. This is accurately reproduced by the proposed model. Several numerical models equipped with special treatment for wave breaking, not often perform very well and fail in reproducing wave breaking in time and/or amplitude [13]. After t(g/h0)1/2 = 20, the front of the wave moves towards the shoreline and the height of the wave dramatically reduces. The wave progressively runs up to the beach and reaches the maximum height around t(g/h0)1/2 = 45. The present model simulates very well the run-up process. During the draw-down, a hydraulic jump is formed approximately at t(g/h0)1/2 = 50. This is accurately simulated by the present model and, generally, good agreement is obtained till the end of the wave retreat process. During the draw-down at t(g/h0)1/2 = 55, several authors [8,13,50] compute numerical results that not perfectly match the experimental data. On the opposite, the proposed model matches very well the measures at t(g/h0)1/2 = 55. Dimakopoulos et al. [51] simulated the same test case using OpenFOAM© [3]. Their results are consistent with the experimental ones. Dimakopoulos et al. [51] discretize the domain with 180000 3D cells and their total simulation time is 10s. They use 12 CPU cores with 2.6 GHz, with a total CPU time of 2 hrs. For the simulations of the present model we use a single processor Intel Q 6600 with 2.4 GHz. The total simulation time is 25 s, requiring 425 s of computational time [49].

Figure 6: Test 3. Measured (circles) and simulated (blue solid line) surface profiles. Measures.

Test 4: Run-up of a solitary wave on a conical island

We deal with the run-up of a tsunami on a conical island [52]. The lab flume (30 m wide and 25 m long, in Figure 7) presents an island, i.e., a truncated circular cone, 0.625 m high, with diameters 7.2 m and 2.2 m at the toe and crest, respectively, and slope of the face 1:4. The surface of the physical model is made of smooth concrete and we assume zero Manning friction coefficient. Other details of the experiments can be found in [52].

Figure 7: Test 4. Definition sketch.

Figure 8: Test 4. Snapshot of the water surface for aw/h0 = 0.181 at different times. The black arrow indicates the direction of the incident wave.

The initial conditions are as in tests 1 and 3, and all the boundaries of the domain are open. The domain is discretized with a GD mesh with 143622 triangles and 72467 nodes. The initial water depth value h0 in our numerical run is 0.32 m and we studied three cases with relative wave heights aw/h0 = 0.045, 0.096 and 0.181, respectively. These experiments have been widely tested in the literature [8,13,17,39,45,53,54].

For brevity, we shown in Figure 8 some snapshots of the wave evolution only for aw/h0 = 0.181, where the dispersive waves are more pronounced, compared with the other two cases. In Figure 8a the wave reaches the maximum height of run-up over the front of the island. After that, the wave withdraws below the initial water level. The refracted waves partially propagate around the island towards the downstream side and two trapped waves are generated at both sides of the cone. These two waves interact at the back face of the island and generate a second run-up. Then, water waves withdraw from the back face of the island and pass through each other, moving around the cone. Figure 8e shows the first group of the dispersive waves wrapping around the island and colliding on the lee side. Figure 8f shows the second group of the dispersive waves after the collision on the back side. Similar observations have been provided in [8,54].

We compare the computed and measured water levels at gauges WG 6, 9, 16 and 22 in Figure 9. The data measured at WG 3 are used for time adjustment of the computed solutions [13]. Generally, the numerical results computed by the present solver match satisfactorily the measured time series. The run-down at the front face of the island, following the first run-up, is properly simulated. The differences between numerical and experimental data become more evident at later times, especially for aw/h0 = 0.181. Fuhrman & Madsen [53] assert that during the experiments, the front of the wave has been generated more accurately than the rear. This could motivate the above discrepancies after the initial run-up. The present model accurately describes the phase and the amplitude of the peak. The peak is slightly delayed at WG 22 for aw/h0 = 0.096 and 0.181 and at WG 9 and 16 in case aw/h0 = 0.181. The amplitude of the peak is reasonably predicted by the present model, which slightly overestimates the amplitude of the leading wave at WG 9 for aw/h0 = 0.096 and 0.181. The depression after peak at WG 9 is slightly underestimated for aw/h0 = 0.096 and overestimated for aw/h0 = 0.181. Other literature models [53,54], over predict the amplitude of the peak at WG 9, 16 and 22, expecially for aw/h0 = 0.18.

Figure 9: Test 4. Measured (circles) and simulated (blue solid line) time evolution of water level at the gauges. Measures from [8]. a w / h 0 = 0.045 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGHbGcpaWaaSbaaKqaGeaajugWa8qacaWG3baal8aabeaa jugib8qacaGGVaGaamiAaOWdamaaBaaajeaibaqcLbmapeGaaGimaa WcpaqabaqcLbsapeGaeyypa0JaaeiiaiaaicdacaGGUaGaaGimaiaa isdacaaI1aaaaa@4533@  (Left panel), a w / h 0 = 0.096 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGHbGcpaWaaSbaaKqaGeaajugWa8qacaWG3baal8aabeaa jugib8qacaGGVaGaamiAaOWdamaaBaaajeaibaqcLbmapeGaaGimaa WcpaqabaqcLbsapeGaeyypa0JaaeiiaOGaaGimaKqzGeGaaiOlaOGa aGimaiaaiMdacaaI2aaaaa@45DC@ (Middle panel), a w / h 0 = 0.181 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGHbGcpaWaaSbaaKqaGeaajugWa8qacaWG3baal8aabeaa jugib8qacaGGVaGaamiAaOWdamaaBaaajeaibaqcLbmapeGaaGimaa WcpaqabaqcLbsapeGaeyypa0JaaeiiaOGaaGimaiaac6cacaaIXaGa aGioaiaaigdaaaa@453E@ (Right panel).

Dimakopoulos et al. [51] applied the open source code OpenFOAM© [3] to reproduce the same test. They simulated half domain, discretized with 6000000 3D cells for a total simulation time 17 sec [51]. Their results are consistent with the experimental ones and very similar to the one provided by the present model. Dimakopoulos et al. [51] used 12 CPU cores with 2.6 GHz, with a total CPU time 24 hrs. The total simulation time in our run is 20 s and, over one processor Intel Q 6600 with 2.4 GHz, the total CPU time is 1795 s.

Test 5: Tsunami in the Monai valley

This benchmark concerns a laboratory scale model of the real 1993 Hokkaido Nansei-Oki Tsunami even, around the Monai Valley. The experimental runs have been performed in the 1:400 scale lab flume of the Central Research Institute for Electric Power Industry (CRIEPI) [55]. Field observations of the real event refer to an initial leading depression wave, usually reproduced in the laboratory by an N-wave [9]. Figure 10, 10.a & 10.b show the bathymetry of the model and the incoming N-wave, respectively [56]. This is assumed as Dirichlet condition assigned at the offshore (western) boundary of the computational domain free slip condition is imposed at the two lateral sides (impervious walls). The GD mesh has 33004 triangles and 16730 nodes and Δt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuiLdqKaamiDaaaa@38C5@  is 0.01 s. The Manning's coefficient n is 0.012 s/m1/3 [9]. The maximum value of the computed CFL number in our simulation is 4.93. Three gauges, located in front of the coast, behind the Muen Island (coordinates (4.521 m, 1.196 m), (4.521 m, 1.696 m) and (4.521 m, 2.196 m), respectively for gauges 1, 2 and 3), have been used for the time registration of the free surface elevation. In Figures 11, 11.a to 11.c shows some snapshots. A series of waves with short period are generated, during the leading depression, by the dispersion of the incoming N-wave over the coastal relief (Figure 11a). The incident wave shoals over the plane slope of the relief and refracts and diffracts around the Muen Island (Figure 11b). After reflection at the coast, some small-amplitude dispersive waves are generated (Figure 11c). At the same time, the waves reflected at the coast interact and generate some breaking waves (Figure 11c). In Figure 12 we compare the measured and the computed elevation of the free surface. The model matches very well the registrations. The initial draw-down, the arrival time and the amplitude of the leading wave are properly reproduced by the model, as well as the reflection at the coast. The discrepancy, approximately for the first 10 s, especially at gauge 1, are due to the initial condition inside the tank of the flume, where η MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4TdGgaaa@3812@  was not uniformly equal to zero [57].

Figure 10: Test 5. a) Bathymetry of the lab model. b) The N-wave assigned at the western side [56].

Figure 11: Test 5. Snapshots of the water levels at different simulation times.

Figure 12: Test 5. Measured (circles) and simulated (red solid line) water levels at the gauges. Measures from [56].

Kesservani & Liang [58] simulated the same benchmark with different numerical solvers: a Discontinuous Galerkin (RKDG2) scheme, an adaptive-grid RKDG2 and a MUSCL2 scheme with 2nd spatial approximation order. They used different resolutions of the spatial discretizations with quadrilateral cells for the RKDG2 scheme [58]. The number of the elements ranged approximately from 7000 to 42000 for the simulation of the adaptive-grid RKDG2, and they used a uniform mesh with 95648 elements for the MUSCL2 scheme [58]. Kesservani & Liang [58] performed their simulations, reproducing a total duration of 25 s, using one processor Core Duo T9400 at 2.53 GHz. The total CPU times required by the models in [58] are listed in Table 4 of the same paper, and range from 0.25 hrs to 8.8 hrs for the RKDG2, 0.75 hrs for the MUSCL2 solver and 1.52 hrs for the adaptive-grid RKDG2. We used one processor Intel Q 6600 at 2.4 GHz and the total computational time was 0.425 hrs for a total simulation time of 200 s. The outputs of the present solver are similar to the results of the adaptive-grid RKDG2 and MUSCL2 schemes in Figure 14 [58], which required higher computational burden. The simulation with the RKDG2 with (98 x 61) elements in [58] obtained CPU times comparable to ours, but returned very bad solutions (Figures 14a-14c).

Conclusions

A non-hydrostatic NLSWEs model has been proposed for the simulations of long waves/tsunamis processes in shallow waters regions. It is our opinion that one of the principal merit of the present solver with respect to others literature schemes proposed for the solution of the same equations [13,19,39,49,] is that in both the hydrostatic and dynamic steps, the computed flow field satisfies the depth-integrated continuity equation discretized in each computational cell. As addressed in the previous sections, this has important implications in the local and global mass balance of the numerical solver. We provide several applications involving breaking/non-breaking waves and run-up, also over irregular topography. Generally, the computed results are in good agreement with the analytical solutions or lab data and with the results provided by other literature algorithms. Another important merit of the proposed model concerns the required computational effort. The CPU times are several magnitude order less than the ones of 3D models and, generally, much less that the ones of other 2D depth-integrated NLSWEs models. All the steps of the proposed algorithm can be parallelizable, as an important future advancement for this study, since it could reduce the computational burden. The parallelization of the hydrostatic CP and CC steps could be obtained by simultaneously solving the ODEs systems in all those cells next to already solved cells with higher and lower values of the approximated potential, respectively. The solution of the systems by the conjugate gradient method, in the hydrostatic DC and dynamic steps, is naturally parallelizable [59]. As a consequence of these considerations, the proposed solver could be embedded in a more general forecasting and prevention tool for the inundation generated by tsunamis or solitary waves.

Disclosure Statement

Some details of the proposed numerical procedure are provided in the file "supplementary material.docx" (Appendices A, B and C), as well as some of the figures and tables concerning the presented test cases (Appendix D).

Acknowledgment

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Conflict of Interest

The Author of the paper, Costanza Aricò, discloses any actual or potential conflict of interest including any financial, personal or other relationships with other people or organizations within three (3) years of beginning the submitted work.

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