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Applied Numerical Mathematics 62 (2012) 1749–1766 Contents lists available at SciVerse ScienceDirect Applied Numerical Mathematics www.elsevier.com/locate/apnum Numerical modelling of sediment transport in the Nador lagoon (Morocco) Fayssal Benkhaldoun a , Salah Daoudi b , Imad Elmahi b,, Mohammed Seaid c a LAGA, Université Paris 13, 99 Av. J.B. Clement, 93430 Villetaneuse, France b ENSAO, EMSN, COSTE, Université Mohammed 1, B.P. 669, 60000 Oujda, Morocco c School of Engineering and Computing Sciences, University of Durham, South Road, Durham DH1 3LE, UK article info abstract Article history: Available online 28 June 2012 Keywords: Nador lagoon Shallow water equations Sediment transport Morphodynamics Finite volume method Unstructured mesh We present a numerical method for solving the sediment transport in the Nador lagoon. The lagoon is located on the Moroccan eastern coast and exchanges water flow with the Mediterranean sea. The governing equations consist of the well-established shallow water system including bathymetric forces, friction terms, coriolis and eddy-diffusion stresses. To model sediment transport we consider an Exner equation for morphological evolution and an advection–diffusion problem for the transport of suspended sediments. As a numerical solver, we apply an adaptive finite volume method using a centred-type discretization for the source terms. The proposed method can handle complex topography using unstructured grids and satisfies the conservation property. Several numerical results are presented to demonstrate the high resolution of the proposed method and to confirm its capability to provide accurate and efficient simulations for sediment transport in the Nador lagoon. © 2012 IMACS. Published by Elsevier B.V. All rights reserved. 1. Introduction The Nador lagoon is located on the Moroccan eastern coast. It is a restricted lagoon of 115 km 2 (25 km by 7.5 km) and with a depth not exceeding 8 m, see Fig. 1. Recently, the Nador lagoon has been the subject of many investigations on water quality, currents, flora, fauna, fishing and aquaculture, see for instance [9,22]. Most of these studies deal with the environmental aspects of the lagoon such as biological and socioeconomic impacts. However, to the best of our knowledge, there are no research studies on the numerical modelling of sediment transport in the Nador lagoon. Needless to mention that numerical studies are essential since they can quantify the interaction between sediment transport and water flow and thereafter can help to understand the evolution of the lagoon morphodynamics. Consequently, this may provide numerical tools to study the physical environment of the lagoon and to assess the development strategy reducing the flood and pollution risks in the lagoon. Certainly, numerical modelling of sediment transport in the Nador lagoon would be less costly than experimental study on the lagoon field. In the current work, the governing equations consist of the well-established shallow water system including bathymetric forces, Coriolis effects, friction terms and eddy-diffusion stresses. To model sediment transport we consider the well-known Exner equation for morphological evolution and an advection–diffusion problem for the suspended sediment accounting for erosion and deposition effects. The coupled hydrodynamical and morphological model forms a hyperbolic system of conservation laws with source terms which are not easy to solve numerically. For instance, numerical simulation of mor- phodynamical changes of the bed in hydraulic systems involve different physical mechanisms propagating within the system * Corresponding author. E-mail address: [email protected] (I. Elmahi). 0168-9274/$36.00 © 2012 IMACS. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.apnum.2012.05.010

Numerical modelling of sediment transport in the Nador lagoon (Morocco)

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Page 1: Numerical modelling of sediment transport in the Nador lagoon (Morocco)

Applied Numerical Mathematics 62 (2012) 1749–1766

Contents lists available at SciVerse ScienceDirect

Applied Numerical Mathematics

www.elsevier.com/locate/apnum

Numerical modelling of sediment transport in the Nador lagoon(Morocco)

Fayssal Benkhaldoun a, Salah Daoudi b, Imad Elmahi b,∗, Mohammed Seaid c

a LAGA, Université Paris 13, 99 Av. J.B. Clement, 93430 Villetaneuse, Franceb ENSAO, EMSN, COSTE, Université Mohammed 1, B.P. 669, 60000 Oujda, Moroccoc School of Engineering and Computing Sciences, University of Durham, South Road, Durham DH1 3LE, UK

a r t i c l e i n f o a b s t r a c t

Article history:Available online 28 June 2012

Keywords:Nador lagoonShallow water equationsSediment transportMorphodynamicsFinite volume methodUnstructured mesh

We present a numerical method for solving the sediment transport in the Nador lagoon.The lagoon is located on the Moroccan eastern coast and exchanges water flow withthe Mediterranean sea. The governing equations consist of the well-established shallowwater system including bathymetric forces, friction terms, coriolis and eddy-diffusionstresses. To model sediment transport we consider an Exner equation for morphologicalevolution and an advection–diffusion problem for the transport of suspended sediments.As a numerical solver, we apply an adaptive finite volume method using a centred-typediscretization for the source terms. The proposed method can handle complex topographyusing unstructured grids and satisfies the conservation property. Several numerical resultsare presented to demonstrate the high resolution of the proposed method and to confirmits capability to provide accurate and efficient simulations for sediment transport in theNador lagoon.

© 2012 IMACS. Published by Elsevier B.V. All rights reserved.

1. Introduction

The Nador lagoon is located on the Moroccan eastern coast. It is a restricted lagoon of 115 km2 (25 km by 7.5 km)and with a depth not exceeding 8 m, see Fig. 1. Recently, the Nador lagoon has been the subject of many investigationson water quality, currents, flora, fauna, fishing and aquaculture, see for instance [9,22]. Most of these studies deal with theenvironmental aspects of the lagoon such as biological and socioeconomic impacts. However, to the best of our knowledge,there are no research studies on the numerical modelling of sediment transport in the Nador lagoon. Needless to mentionthat numerical studies are essential since they can quantify the interaction between sediment transport and water flow andthereafter can help to understand the evolution of the lagoon morphodynamics. Consequently, this may provide numericaltools to study the physical environment of the lagoon and to assess the development strategy reducing the flood andpollution risks in the lagoon. Certainly, numerical modelling of sediment transport in the Nador lagoon would be less costlythan experimental study on the lagoon field.

In the current work, the governing equations consist of the well-established shallow water system including bathymetricforces, Coriolis effects, friction terms and eddy-diffusion stresses. To model sediment transport we consider the well-knownExner equation for morphological evolution and an advection–diffusion problem for the suspended sediment accountingfor erosion and deposition effects. The coupled hydrodynamical and morphological model forms a hyperbolic system ofconservation laws with source terms which are not easy to solve numerically. For instance, numerical simulation of mor-phodynamical changes of the bed in hydraulic systems involve different physical mechanisms propagating within the system

* Corresponding author.E-mail address: [email protected] (I. Elmahi).

0168-9274/$36.00 © 2012 IMACS. Published by Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.apnum.2012.05.010

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1750 F. Benkhaldoun et al. / Applied Numerical Mathematics 62 (2012) 1749–1766

Fig. 1. Location and schematic description of the Nador lagoon.

according to their time response i.e., the problem is a multi-scale system which requires a robust solver to accurately re-solve both hydrodynamical and morphodynamical time scales. In addition, most morphodynamical flows frequently involveimportant features such as internal bores and moving shoreline fronts which present a significant challenge to the accuracyand stability of numerical models. The objective of this work is to devise a stable, reliable and accurate numerical methodable to approximate solutions to the coupled hydrodynamical and morphological system in the Nador lagoon. It should bestressed that it is difficult to validate numerical results for sediment transport against measurements from field experiments.The main difficulties lie essentially on the empirical formulae needed to close the sediment transport model and also onthe calibration of the parameters involved in the bed-load modelling, we refer the reader to [11] for more discussions.

An adaptive finite volume Non-Homogeneous Riemann Solver (SRNH) has recently been proposed in [6,7] for solving sed-iment transport by shallow water equations without accounting for suspended sediment. Here, the acronym SRNH stands for“Solveur de Riemann Non Homogène”. The SRNH method belongs to the class of methods that employ only physical fluxesand averaged states in their formulations. To control the local diffusion in the scheme and also to preserve monotonicity, aparameter is introduced based on the sign matrix of the flux Jacobian. The main features of such a SRNH scheme are: (i) thecapability to satisfy the conservation property resulting in numerical solutions free from spurious oscillations in significantmorphodynamic situations (ii) the implementation on unstructured meshes allowing for local mesh refinement during thesimulation process, and (iii) the achievement of strong stability for simulations of slowly varying bed-load as well as rapidlyvarying flows containing also shocks or discontinuities. It should also be stressed that the SRNH method has been success-fully applied to solve pollutant dispersion by shallow water flows in [3]. Neither eddy diffusivity nor wind shear stressesat the water surface have been taken into account in the study presented in [3]. Nevertheless, our SRNH method has beenfound to perform effectively on realistic irregular bottoms, handle complex geometry, to solve different hydraulic regimes,and to preserve conservation properties for both water free-surface and pollutant concentration. Our aim in this paper is toextend the SRNH method for solving the problem of sediment transport in the Nador lagoon. We should mention that in thecurrent work, we are not interested in solutions with shocks within the Nador lagoon. Numerical results presented in thisstudy show that an interesting feature of the SRNH method is to allow multilevel mesh adaptation without deterioratingaccuracy of the computed solutions.

The structure of this paper is as follows. In Section 2, we present the mathematical equations for the sediment transportproblems considered. The formulation of the adaptive finite volume method is detailed in Section 3. Section 4 is devoted tonumerical results. Finally, Section 5 contains the conclusions.

2. Governing equations for sediment transport problems

In this section we describe the physical model used for modelling the sediment transport by shallow water flows. Here,the two-dimensional shallow water equations are briefly recast for the hydrodynamics followed with a short description forthe sediment transport equations for the morphodynamics.

2.1. Shallow water equations

Shallow water equations have been widely used to model free surface flows of a fluid under the influence of gravity. Thisclass of equations uses the assumption that the vertical scale is much smaller than any typical horizontal scale and can be

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F. Benkhaldoun et al. / Applied Numerical Mathematics 62 (2012) 1749–1766 1751

Fig. 2. Sketch of a domain for two-dimensional suspended sediment transport.

derived from the depth-averaged incompressible Navier–Stokes equations, compare [23] among others. For two-dimensionalflow problems, these equations are:

∂h

∂t+ ∂(hu)

∂x+ ∂(hv)

∂ y= 0, (1a)

∂(hu)

∂t+ ∂(hu2 + 1

2 gh2)

∂x+ ∂(huv)

∂ y= −gh

∂ B

∂x− Ωhv − τbx

ρw+ τwx

ρw+Dxx(h, u, v), (1b)

∂(hv)

∂t+ ∂(huv)

∂x+ ∂(hv2 + 1

2 gh2)

∂ y= −gh

∂ B

∂ y+ Ωhu − τby

ρw+ τwy

ρw+Dyy(h, u, v), (1c)

where u and v are the depth-averaged water velocities in x- and y-direction, h the water depth, B the bottom topog-raphy, g the gravitational acceleration, ρw the water density, Ω the Coriolis parameter defined by Ω = 2ω sin φ, withω = 0.000073 rad s−1 is the angular velocity of the earth and φ is the geographic latitude, see Fig. 2 for an illustration.Here, τbx and τby the bed shear stress in the x- and y-direction, respectively, defined by the depth-averaged velocities as

τbx = ρw Cbu√

u2 + v2, τby = ρw Cb v√

u2 + v2, (1d)

where Cb is the bed friction coefficient, which may be either constant or estimated as Cb = g/C2z , where Cz = h1/6/nb is the

Chezy constant, with nb being the Manning roughness coefficient at the bed. The surface stress τw is usually originated bythe shear of the blowing wind and is expressed as a quadratic function of the wind velocity,

τwx = ρw C w wx

√w2

x + w2y, τby = ρw C w w y

√w2

x + w2y, (1e)

with C w is the coefficient of wind friction and w = (wx, w y)T is the velocity of wind at 10 m above water surface. It is

usually defined by

C w = ρa

(0.75 + 0.067

√w2

x + w2y

)× 10−3,

where ρa is the air density. For the diffusion terms Dxx(h, u, v) and Dyy(h, u, v) we have adopted the model derived in[12,16] as

Dxx(h, u, v) = 2ν∂

∂x

(h

(2∂u

∂x+ ∂v

∂ y

))+ 2ν

∂ y

(h

∂u

∂ y

),

Dyy(h, u, v) = 2ν∂

∂x

(h∂v

∂x

)+ 2ν

∂ y

(h

(∂u

∂x+ 2

∂v

∂ y

)),

where ν is the kinematic viscosity. Note that we have assumed water flow at laminar regime however, turbulent effects canalso be accounted for in the model (1) by modifying the eddy viscosity coefficients. In addition, other coefficients of windfriction can also be applied in (1e).

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1752 F. Benkhaldoun et al. / Applied Numerical Mathematics 62 (2012) 1749–1766

2.2. Suspended sediment equations

Most of existing formulations of sediment transport are empirical and have been derived based on experimental dataand measurements. There are many mathematical models for sediment transport by shallow water flows. In the currentwork, we consider the advection–diffusion equation

∂(hC)

∂t+ ∂(huC)

∂x+ ∂(hvC)

∂ y= ∂

∂x

(κh

∂C

∂x

)+ ∂

∂ y

(κh

∂C

∂ y

)+ E − D + hQ , (2a)

where C is the depth-averaged concentration of suspended sediment, Q is the depth-averaged sediment source, and κ isthe diffusion coefficient of the sediment. In Eq. (2a), E and D represent the erosion and deposition terms in upward anddownward directions, respectively. In the current study, the erosion–deposition term is calculated as

E − D = αωs(C∗ − C), (2b)

where ωs is the settling velocity of sediment particles, α is the recovery coefficient of the suspended sediment and C∗ isthe sediment concentration close to the bed defined by [20]

C∗ = 0.015d50Tδbd0.3∗

, (2c)

with d50 is the mean diameter of the sediment and δb is a reference level fixed in our computations to 0.05h, compare [18].The excess bed shear stress T is defined as

T = u2∗ − u2c

u2c

, (2d)

where the shear velocity u∗ and the critical bed shear velocity uc for the sediment are given by

u2∗ =√

τ 2bx + τ 2

by

ρw, uc =

√a

(d50

10

)b

,

with a and b are constants depending on the mean diameter of the sediment particles. In (2c), the particle-size diameter d∗is defined as

d∗ = d50

(gρs − ρw

ρwν2

) 13

, (2e)

where ρs is the sediment density. Note that the parameters α, ρs , a, and b are user-defined constants in the sedimenttransport model.

2.3. Bed-load equations

In (1a), the function B corresponds to the sediment layer characterizing the bed level. For fixed bottom topography,i.e., B = B(x), Eq. (1a) reduces to the standard shallow water equations. In the current work, we assume that a sedimenttransport takes place such that the bed level depends on the time variable as well. This requires an additional equation forits evolution. Here, to update the bed-load, we use the Exner equation given by

(1 − p)∂ B

∂t+ ∂ Q bx

∂x+ ∂ Q by

∂ y= ∂

∂x

∂ B

∂x

)+ ∂

∂ y

∂ B

∂ y

), (3a)

where p is the sediment porosity assumed to be constant and ζ is the diffusion coefficient. In Eq. (3a), Q bx and Q byrepresent the bed-load sediment transport fluxes in x- and y-direction, respectively. These fluxes depend on the type ofsediment and for simplicity in the presentation, we consider the basic sediment transport fluxes proposed by Grass in [13]

Q bx = Au(u2 + v2)m−1

2 , Q by = Av(u2 + v2)m−1

2 , (3b)

where A is a given experimental constant and 1 � m � 4 is a chosen parameter both of which are specific to the particularsediment transport formula. In all simulations presented in this paper, the parameter m = 3. It should be stressed that thefinite volume method described in this paper can be applied to other forms of sediment transport fluxes without majorconceptual modifications. For instance, the bed-load sediment transport functions proposed in [19,21] can also be handledby the proposed finite volume method. Notice that, in practical situations the diffusion coefficients ν , κ and ζ depend onwater depth, flow velocity, bottom roughness, wind and vertical turbulence, compare [15] for more details. For the purposeof the present work, the problem of the evaluation of diffusion coefficients is not considered.

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F. Benkhaldoun et al. / Applied Numerical Mathematics 62 (2012) 1749–1766 1753

Fig. 3. Generic control volume and notation.

For simplicity in presentation we can also reformulate Eqs. (1), (2) and (3) in a compact conservative form as

∂W

∂t+ ∂

∂x

(F(W) − F(W)

) + ∂

∂ y

(G(W) − G(W)

) = Q(W) + S(W), (4)

where W is the vector of conserved variables, Q is the source term accounting for the slope variations, S is the source termaccounting for Coriolis forces, friction losses and erosion–deposition terms, F and G are the advective tensor fluxes, F and Gare the diffusion tensor fluxes, compare [3] for a similar formulation for shallow water equations over fixed beds. In (4)

W =

⎛⎜⎜⎜⎝h

huhvhCB

⎞⎟⎟⎟⎠ ,

Q(W) =

⎛⎜⎜⎜⎜⎜⎝0

−gh ∂ B∂x

−gh ∂ B∂ y

00

⎞⎟⎟⎟⎟⎟⎠ , S(W) =

⎛⎜⎜⎜⎜⎜⎝0

−Ωhv − τbxρw

+ τwxρw

Ωhu − τbyρw

+ τwyρw

E − D + hQ0

⎞⎟⎟⎟⎟⎟⎠ , (5)

F(W) =

⎛⎜⎜⎜⎜⎝hu

hu2 + 12 gh2

huvhuCQ bx1−p

⎞⎟⎟⎟⎟⎠ , G(W) =

⎛⎜⎜⎜⎜⎝hv

huvhv2 + 1

2 gh2

hvCQ by1−p

⎞⎟⎟⎟⎟⎠ ,

F(W) =

⎛⎜⎜⎜⎜⎜⎜⎝

02νh(2 ∂u

∂x + ∂v∂ y )

2νh ∂v∂x

κh ∂C∂x

ζ1−p

∂ B∂x

⎞⎟⎟⎟⎟⎟⎟⎠ , G(W) =

⎛⎜⎜⎜⎜⎜⎝0

2νh ∂u∂ y

2νh( ∂u∂x + 2 ∂v

∂ y )

κh ∂C∂ y

ζ1−p

∂ B∂ y

⎞⎟⎟⎟⎟⎟⎠ .

Note that we have considered only a single suspended sediment with concentration C transported by the shallow waterflow, however the techniques presented in this paper can straightforwardly be extended to sediment transport with multiplespecies.

3. Finite volume non-homogeneous Riemann solver

The main advantages of the finite volume methods lie on their implementation on unstructured triangular meshes andpreserving conservation properties of the equations. Hence, using the control volume depicted in Fig. 3, a finite volumediscretization of (4) yields

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1754 F. Benkhaldoun et al. / Applied Numerical Mathematics 62 (2012) 1749–1766

Wn+1i = Wn

i − t

|Ti|∑

j∈N(i)

∫Γi j

F(Wn;η)

dσ + t

|Ti|∑

j∈N(i)

∫Γi j

F(Wn;η)

+ t

|Ti|∫Ti

Q(Wn)dV + t

|Ti|∫Ti

S(Wn)dV , (6)

where N(i) is the set of neighboring triangles of the cell Ti , Wni is an average value of the solution W in the cell Ti at

time tn ,

Wi = 1

|Ti|∫Ti

W dV ,

where |Ti | denotes the area of Ti and Si is the surface surrounding the control volume Ti . Here, Γi j is the interface betweenthe two control volumes Ti and T j , η = (nx,ny)

T denotes the unit outward normal to the surface Si , and

F(W;η) = F(W)nx + G(W)ny, F(W;η) = F(W)nx + G(W)ny .

Note that the time stepping in (6) is only first-order accurate. A second-order accuracy in time can be achieved by consider-ing a two-step Runge–Kutta method. Its implementation for solving the semi-discrete equations (6) is straightforward. Thefinite volume discretization (6) is complete once the gradient fluxes F(Wn;η), diffusion fluxes F(Wn;η) and source termsQ(Wn) and S(Wn) are reconstructed.

3.1. Discretization of gradient fluxes

Let us consider the hyperbolic part in the system (4)

∂W

∂t+ ∂F(W)

∂x+ ∂G(W)

∂ y= Q(W), (7)

where the source term Q accounts only of the bed slopes given in (5). Applied to the system (7), the finite volume dis-cretization (6) yields

∂t

∫Ti

h dV +∮Si

(hunx + hvny)dσ = 0,

∂t

∫Ti

hu dV +∮Si

((hu2 + 1

2gh2

)nx + huvny

)dσ = −gh

∮Si

Bnx dσ ,

∂t

∫Ti

hv dV +∮Si

(huvnx +

(hv2 + 1

2gh2

)ny

)dσ = −gh

∮Si

Bny dσ ,

∂t

∫Ti

hC dV +∮Si

(huCnx + hvCny)dσ = 0,

∂t

∫Ti

B dV + Aξ

∮Si

(u(u2 + v2)nx + v

(u2 + v2)ny

)dσ = 0,

where ξ = 1/(1− p). Using the local cell outward normal η and tangential τ = η⊥ , the above equations can be reformulatedas

∂t

∫Ti

h dV +∮Si

huη dσ = 0,

∂t

∫Ti

hu dV +∮Si

(huuη + 1

2gh2nx

)dσ = −gh

∮Si

Bnx dσ ,

∂t

∫hv dV +

∮ (hvuη + 1

2gh2ny

)dσ = −gh

∮Bny dσ ,

Ti Si Si

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F. Benkhaldoun et al. / Applied Numerical Mathematics 62 (2012) 1749–1766 1755

∂t

∫Ti

hC dV +∮Si

huηCdσ = 0,

∂t

∫Ti

B dV + Aξ

∮Si

(u2 + v2)uη dσ = 0, (8)

where the normal velocity uη = unx + vny and tangential velocity uτ = uny − vnx . In order to simplify the system (8), wefirst sum the second equation multiplied by nx to the third equation multiplied by ny , then we subtract the third equationmultiplied by nx from the second equation multiplied by ny . The result of these operations is

∂t

∫Ti

h dV +∮Si

huη dσ = 0,

∂t

∫Ti

huη dV +∮Si

(huηuη + 1

2gh2

)dσ = −gh

∮Si

B dσ ,

∂t

∫Ti

huτ dV +∮Si

huτ uη dσ = 0,

∂t

∫Ti

hC dV +∮Si

huηCdσ = 0,

∂t

∫Ti

B dV + Aξ

∮Si

(u2 + v2)uη dσ = 0, (9)

which can be reformulated in differential form as

∂h

∂t+ ∂(huη)

∂η= 0,

∂(huη)

∂t+ ∂

∂η

(hu2

η + 1

2gh2

)= −gh

∂ B

∂η,

∂(huτ )

∂t+ ∂

∂η(huηuτ ) = 0,

∂(hC)

∂t+ ∂

∂η(huηC) = 0,

∂ B

∂t+ ∂

∂η

(Aξuη

(u2

η + u2τ

)) = 0. (10)

Hence, the system (10) can also be rewritten in a vector form as

∂W

∂t+ Aη(W)

∂W

∂η= 0, (11)

where

W =

⎛⎜⎜⎜⎝h

huη

huτ

hCB

⎞⎟⎟⎟⎠ ,

Aη(W) =

⎛⎜⎜⎜⎜⎝0 1 0 0 0

gh − u2η 2uη 0 0 gh

−uηuτ uτ uη 0 0−uηC C 0 uη 0

−Aξ3uη(u2

η+u2τ )

h Aξ3 u2

η+u2τ

h Aξ2uηuτ

h 0 0

⎞⎟⎟⎟⎟⎠ .

Note that one of the advantages in considering the projected system (11) is that no discretization of source terms is required.Thus, in the predictor stage, we use the projected system (11) to compute the averaged states as

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1756 F. Benkhaldoun et al. / Applied Numerical Mathematics 62 (2012) 1749–1766

Wni j = 1

2

(Wn

i + Wnj

) − 1

2sgn

[Aη(W)

](Wn

j − Wni

), (12)

where the sign matrix is defined as

sgn[Aη(W)

] = R(W) sgn[Λ(W)

]R−1(W),

with Λ(W) is the diagonal matrix of eigenvalues, R(W) is the right eigenvector matrix and W is the Roe’s average stategiven by

W =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

hi+h j2

hi+h j2 (

ui√

hi+u j√

h j√hi+

√h j

ηx + vi√

hi+v j√

h j√hi+

√h j

ηy)

hi+h j2 (− ui

√hi+u j

√h j√

hi+√

h jηy + vi

√hi+v j

√h j√

hi+√

h jηx)

hi+h j2 (

Ci√

hi+C j√

h j√hi+

√h j

)

Bi+B j2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (13)

The matrices Λ(W) and R(W) can be explicitly expressed using the associated eigenvalues of Aη(W). For convenience ofthe reader, these matrices are formulated in Appendix A.

Once the projected states are calculated in the predictor stage (12), the states Wni j are recovered by using the trans-

formations v = (uτ , uη) · η and u = (uτ , uη) · τ . Thus, applied to the system (7), the proposed SRNH scheme consists of apredictor stage and a corrector stage and can be formulated as

Wni j = 1

2

(Wn

i + Wnj

) − 1

2sgn

[Aη(W)

](Wn

j − Wni

),

Wn+1i = Wn

i − t

|Ti|∑

j∈N(i)

F(Wn

i j;ηi j)|Γi j| + tQn

i , (14)

where Qni is a consistent discretization of the source term in (7) defined as

Qni =

⎛⎜⎜⎜⎜⎝0

−ghnxi

∑j∈N(i) Bijnxi j|Γi j|

−ghnyi

∑j∈N(i) Bijnyi j|Γi j|

00

⎞⎟⎟⎟⎟⎠ . (15)

The approximations hnxi and hn

yi are reconstructed in such a way the SRNH scheme satisfies the well-known C-property, seefor example [8,3,6,7]. Thus,

hnxi = 1

2|Ti|

∑j∈N(i)(h

nij)

2nxi j|Γi j|∑j∈N(i) hn

ijnxi j|Γi j| , hnyi = 1

2|Ti|

∑j∈N(i)(h

nij)

2nyij|Γi j|∑j∈N(i) hn

ijnyi j|Γi j| . (16)

Note that we have used a quadrature rule to approximate the integrals in the projected system (8). A more detailed for-mulation on how these integrals are evaluated, by solving a linear system within each control volume, can be found in [5].Obviously, the finite volume method (14) is only first-order accurate. In order to develop a second-order finite volumescheme, we use a MUSCL method incorporating slope limiters in the spatial approximation. The MUSCL discretization usesan approximation of the solution state W by linear interpolation at each cell interface Γi j as

Wi j = Wi + 1

2∇Wi · di j, W ji = W j − 1

2∇W j · di j, (17)

where xi = (xi, yi)T and x j = (x j, y j)

T are respectively, the barycentric coordinates of cells Ti and T j , and di j is the distancebetween xi and x j , compare Fig. 3. Thus, the cell gradients are evaluated by minimizing the quadratic functional

Ψi(x, y) =∑

j∈M(i)

∣∣Wi + (x j − xi)x + (y j − yi)y − W j∣∣2

, (18)

where M(i) is the set of indices of neighboring cells that have a common edge or vertex with the control volume Ti . Forinstance, ∇Wi = (

∂Wi∂x ,

∂Wi∂ y )T in (17) are solutions of the linear system

∂Ψi(x, y) = 0,∂Ψi(x, y) = 0.

∂x ∂ y

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It is easy to verify that

∂Wi

∂x= J x I yy − J y Ixy

D,

∂Wi

∂ y= J y Ixx − J x I yx

D, (19)

where D = Ixx I yy − Ixy I yx and

Ixx =∑

j∈M(i)

(x j − xi)2, I yy =

∑j∈M(i)

(y j − yi)2,

Ixy = I yx =∑

j∈M(i)

(x j − xi)(y j − yi),

J x =∑

j∈M(i)

(x j − xi)(W j − Wi), J y =∑

j∈M(i)

(y j − yi)(W j − Wi).

In order to obtain a TVD scheme, we incorporate slope limiters to the reconstruction (17)–(18) using the MinMod limiterfunction. This is achieved by replacing the cell gradients in (19) by

∂ limWi

∂x= 1

2

(min

j∈M(i)sgn

[∂W j

∂x

]+ max

j∈M(i)sgn

[∂W j

∂x

])min

j∈M(i)

∣∣∣∣∂W j

∂x

∣∣∣∣, (20)

with a similar expression for ∂ limWi∂ y . It should be pointed out that this slope limiter function is very easy to implement,

but it may cause some numerical smoothing of the solution, we refer to [3] for further discussions. More sophisticatedlimiters that are less dissipative are currently under investigation for sediment transport problems. It should be pointed outthat, in order to achieve a second-order accuracy in the proposed SRNH scheme, the state solutions Wi j and W ji in thediscretization of gradient fluxes and source terms should be replaced by the second-order reconstruction (17).

3.2. Discretization of diffusion terms

To discretize the diffusion fluxes in (6) we adapt a Green-Gauss diamond reconstruction, see for example [10,3] andfurther references are therein. This method has been selected because it is second-order accurate, it can be applied ongeneral unstructured adaptive grids, it does not require serious restrictions on the angles of triangles, and it can be easilyincorporated in our SRNH scheme. Hence, a co-volume coVij is first constructed by connecting the barycentres of theelements that share the edge Γi j and its endpoints as shown in Fig. 3. Then, in the x-direction, diffusion fluxes in thesediment transport equation are evaluated at an inner edge Γi j as∫

Γi j

κh∂C

∂xnx dσ = κh|Γi j

|coVij|∑

ε∈∂coVij

CN1 + CN2

2

∫ε

nxε dσ , (21)

where N1 and N2 are the nodes of the edge ε on the surface ∂coVij , CN1 and CN2 are the values of the concentration Cin the node N1 and N2, respectively. The discretization in y-direction of the diffusion fluxes is carried out in an analogousmanner. Note that, in (21) we have assumed constant diffusion coefficient. In the case of space dependent diffusion, thecoefficient κ in (21) should be replaced by

κN1 + κN2 + κN3 + κN4

4,

with κNk , k = 1, . . . ,4, are values of the diffusion coefficient κ at the co-volume nodes Nk approximated by linear interpo-lation from the values on the cells sharing the same vertex Nk .

Finally, to discretize the friction, Coriolis and erosion–deposition terms contained in the source term S(W) in (5), weconsider an operator splitting procedure. Thus, the system (4) is decomposed in two equations as

∂W

∂t+ Res(W) = Q(W),

∂W

∂t= S(W), (22)

where Res describes the convection and diffusion terms in the momentum equation corresponding to the surface integralin (4) and it is approximated as the sum taken over all edges of each element in the computational mesh.

First, an explicit method is used to integrate the first equation in (22) leading to

W − Wn

t+ Res

(Wn) = Q

(Wn). (23)

In the second step, the state solution W is taken to be the initial condition when solving the second equation in (22).

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Fig. 4. Illustration of a two-level refining. The numbers in the figure refer to the number of refinements to be performed for each subtriangle.

3.3. Procedure for mesh adaptation

The mesh generation is based on the Delaunay triangulation [14], which uses a curvature-dependent generation strategydesigned to produce smaller elements in regions of high curvature in the spatial domain. In order to improve the efficiencyof the SRNH scheme, we have performed a mesh adaptation to construct a nearly optimal mesh able to capture the smallhydraulic and sediment features without relying on extremely fine grid in smooth regions far from concentration or hy-draulic jumps. In the present work, this goal is achieved by using an error indicator for the concentration of the suspendedsediment. This indicator requires only information from solution values within a single element at a time and it is easilycalculated. Other adaptation techniques based on the estimation of gradients such as those studied in [1] can also be ap-plied. However, these error estimations can be computationally very demanding since a global solution step is needed toproject the gradients on a linear basis.

The adaptive procedure used here is based on multilevel refinement and unrefinement, it is aimed at constructing anadaptive mesh which dynamically follows the unsteady solution of the physical problem. This procedure has been usedrecently in [10] for adaptive finite volume solution of a combustion system and in [4] for pollutant transport by shallowwater flows. The algorithm begins by selecting some criterion (here based on the concentration of the suspended sediment),which permits to make the refinement and unrefinement decisions. A list S of elements to be refined, their degree ofrefinement, and those to be unrefined is then established. This is accomplished by filling an integer array denoted forexample by I for all triangles of the coarse mesh. At time t = tn and for a macro-element Ti we set I(Ti) = m whichmeans that the element Ti has to be divided into 4m triangles. Thus, starting from a mesh level l, made of N(l) cells, thenext mesh level will contain N(l+1) = 4 × N(l) cells. Clearly, this process can be repeated as long as l < mmax with mmaxbeing the number of refinement levels. In order to obtain a mesh which is not too distorted, the algorithm decides to divideinto two equal parts some additional edges. An illustration of the adaptation procedure is shown in Fig. 4.

In our simulations, the adaption criterion is based on the normalized concentration of the suspended sediment, and isevaluated as

Critn(Ti) = C(Ti)

maxT j C(T j), (24)

where C(Ti) is the sediment concentration on cell Ti . The advantage of this normalization is that the criterion (24) is knownto take its values in the interval [0,1]. Hence, an adaptation procedure can be performed as follows:

Given a sequence of three real numbers {rm} such that 0 = r0 < r1 < r2 < r3 < r4 = 1. If a macro-element Ti satisfies thecondition

rm � Critn(Ti) � rm+1, m = 0, . . . ,3,

then Ti is divided into 4m triangles. Note that the values of {r1, r2, r3} can be interpreted as tolerances to be set by the userresulting into a two-level refining.

4. Numerical results

We consider the limited coastal region on the Nador lagoon shown in the left plot in Fig. 1. The coastal boundary and thebed surface topography of the Nador lagoon are very irregular and several regions of various depths coexist in the lagoonwith the minimum bathymetric values of 9 m and 8 m are localized in the center of the lagoon. As an initial bed bottomwe used the reconstructed bathymetry illustrated in Fig. 5. It should be pointed out that values of the topography werecalibrated to cover all the unstructured meshes used at each level of refinement. Certainly, this will add some computationaleffort to the adaptive SRNH scheme. The selected values for the evaluation of the present model are summarized in Table 1.The model is started from rest and a well-developed discharge of 10 m2/s is imposed at the entrance of the lagoon. Thisdischarge corresponds to the annual mean of the Mediterranean input flux and it is also comparable to the flow generatedby the main semi-diurnal M2 tide in the lagoon. For the suspended sediment we assume a small hump to be localized nearthe entrance of the lagoon with a maximum concentration of order 1. In this sense, the simulations are schematic, since the

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Fig. 5. Reconstructed initial bottom bed used for simulations in the Nador lagoon.

Table 1Parameters and reference quantities considered in the present study.

Quantity Reference value

ρw 1000 kg/m3

ρs 2600 kg/m3

ρa 1200 kg/m3

g 9.81 m/s2

ν 10−6 m2/sκ 10−6 m2/sζ 10−6 m2/snb 0.0157 s/m1/3

Quantity Reference value

A 10−4

d50 1 mmf 8.55 × 10−5/sα 0.2ws 0.0237p 0.4a 0.0055b 1

number, the arrangement, and the capacities of suspended sediment in the Nador lagoon only partially correspond to thereal situation. All the results presented in this section are computed using two-level refining with variable time stepsizes t adjusted at each step according to the CFL condition

t = c min( tconv, tdiff),

where

tconv = minΓi j

( |Ti| + |T j|2|Γi j|maxp |(λp)i j|

), tdiff = min

Γi j

( |Ti|2 max(ν,κ, ζ )

),

with Γi j is the edge between two cells Ti and T j , and c is the Courant number set to 0.8 for all test cases to ensure stability.

In Fig. 6 we present the computed results for a test example without the erosion–deposition effects (i.e., E − D = 0 in(2b)) using the adaptive SRNH scheme at three different times t = 1.3, 3 and 5 hours corresponding to the time requiredby the sediment to evolve towards the south if no wind effects are taken into account. In this figure we show the adaptedmeshes, velocity fields and contours of the normalized concentration, C/Cmax, where Cmax denotes the maximal value ofthe initial concentration of the suspended sediment. Notice that no adaption to the bathymetry has been performed in thistest case. It is clear that the adaptive mesh procedure was able to capture the complex features of the flow with a highlevel of accuracy. The results also show that the adaptive SRNH scheme was able to predict complex wave interactionswith accuracy and to capture sediment concentration with sharp resolution. By using an adaptive grid, high resolution isautomatically obtained in those regions where the sediment concentration gradients are steep such as the moving fronts. Forthe conditions considered, the adaptive SRNH scheme gives a shock-capturing method with very little numerical dissipation,even after long time simulations are carried out.

The results obtained for the test example with the erosion–deposition effects (i.e., E − D �= 0) are depicted in Fig. 7. Fromthe computed results we can also observe that, for the considered flow and sediment conditions, the suspended sedimentis transported faster than the previous case. The dispersion of the sediment is also larger than the case without erosion–deposition terms. During its advection, the sediment alerts the flow structure developing recirculation zones with differentorder of magnitude in the middle and the south coast of the lagoon. The downstream recirculation zone near the southcoast is mainly due to the concentration differences at the region near the exit boundary. In summary, the transport ofsediment is captured accurately, the flow field is resolved reasonably well, and the fronts are shape preserving. All thesefeatures have been achieved using adaptive techniques on unstructured meshes. The presented results demonstrate that

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n–deposition at three different simulation times using the

Fig. 6. Adapted meshes (first column), velocity vectors (second column) and sediment concentration (third column) for sediment transport without erosioGrass formula. From top to bottom times t = 1.3, 3 and 5 hours.
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on–deposition at three different simulation times using the

Fig. 7. Adapted meshes (first column), velocity vectors (second column) and sediment concentration (third column) for sediment transport with erosiGrass formula. From top to bottom t = 1.3, 3 and 5 hours.
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Fig. 8. Evolution in time of sediment concentration at three different gauges G1, G2 and G3 in the Nador lagoon using the Grass formula.

Table 2Mesh statistics and computational times for the considered test examples with and without erosion–deposition effects. The CPU times are given in minutes.

Without erosion–deposition With erosion–deposition

t = 1.3 h t = 3 h t = 5 h t = 1.3 h t = 3 h t = 5 h

# of elements 12 947 12 833 12 115 16 017 25 910 45 292# of nodes 6978 6921 6562 8552 13 504 23 295CPU time 26 62 110 45 102 210

the proposed method is suited for the prediction of coupled hydrodynamics and morphodynamics in the Nador lagoon. Itshould be stressed that results from the proposed sediment transport model should be compared with observations of realsea-surface flow in the Nador lagoon. However, there is no data available until now to carry out this work. Thus, at themoment we can only perform simulations and verify that results are plausible and consistent.

The SRNH method performs very satisfactorily on this nonlinear coupled problem since it does not diffuse the movingsediment fronts and no spurious oscillations have been detected near steep gradients of the flow field in the computationaldomain. It can be clearly seen that the erosion–deposition terms plays an important role in the sediment transport andchanges not only the transport behaviour but also the flow features. In both simulations, the flow structures on the Nadorlagoon are being well captured by the SRNH method. To further quantify the resolution of the SRNH method, we displayin Fig. 8 the evolution in time of the sediment concentration at three relevant locations G1, G2 and G3 situated in theNador lagoon. The coordinates of the gauges G1, G2 and G3 are (9 km, 12.4 km), (7 km, 12.4 km) and (3.6 km, 11.4 km),respectively. It is clear that the simulated results predict the correct sediment trends at each gauge according on how farthe gauge is from the lagoon entry. Note that, since physical diffusion is incorporated into the sediment transport model,the maximum concentration of the suspended sediment was reduced in the two test examples considered with or withouterosion–deposition effects.

Next, analysis of computational cost has been carried out for the test cases with and without erosion–deposition effects.Table 2 summarizes the mesh statistics and computational cost of the SRNH method for the presented simulations at timest = 1.3, 3 and 5 hours. All the computations were performed on the cluster machines of LAGA in Paris 13 university. Themain features reported in this table are on one hand, accounting for erosion–deposition effects in the sediment transportmodel results in an increase in the computational time compared to the CPU time used by the simulation without erosion–deposition terms. On the other hand, the number of elements in the final adaptive mesh decreases with the simulation timefor the test example without erosion–deposition terms and increases for the test example with erosion–deposition terms.This can be attributed to the fact that, for the simulations with erosion–deposition effects the sediment dispersion coversa larger area in the lagoon compared to the sediment advection in simulations without erosion–deposition terms. Theseresults offer a clear impression on how sensitive the numerical prediction of morphological response of the suspended sed-iment. It is noteworthy that, the simulations presented in this study show that large scale sediment transport models withirregular topography can be efficiently solved using personal computers without relying on complicated supercomputingresources.

Finally, to examine the dependence of the numerical results on the selection of the sediment discharge Q b , we comparethe results obtained using the Grass formula (3b) to those obtained using the Meyer-Peter and Müller (MPM) formula givenin [17] by

Q bx = Apu

u2 + v2max(T∗ − Tc,0)

32 ,

Q by = Apv

2 2max(T∗ − Tc,0)

32 , (25)

u + v

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where A p = 8√

g(s − 1)d350 and

T∗ = n2b

√u2 + v2

(s − 1)d50h13

,

with d50 is the median diameter of the sediment and the grain specific gravity s = ρsρw

, with ρs is the sediment density.In our simulations, we solve the bed-load equation (3a) subject to the sediment fluxes (25) using Tc = 0.047 and theremaining parameters are given in Table 1. Fig. 9 presents the results obtained for the test example with the erosion–deposition effects. A simple inspection of these results and those depicted in Fig. 7 reveals that, for the considered set ofparameters in both formulae, there is no significant differences between the results obtained using the Grass model andthe Meyer-Peter and Müller model. Comparison of time evolution of the sediment concentration at the gauge G2 obtainedfor both models shows no noticeable differences in the results by Grass model and the Meyer-Peter and Müller model(see Fig. 10). Similar trends have been observed in other results not reported here for the case of sediment transportwithout erosion–deposition. This may be explained by the fact that sediment transport in the Nador lagoon is controlledby the suspended sediments rather than the morphodynamics of the lagoon bathymetry. It is worth stressing that formorphodynamic-dominant sediment transport problems the Meyer-Peter and Müller model resolves the problem underconsideration with less dissipation than the Grass model, we refer the reader to [2] for an extensive comparative study ofthese morphodynamic models in contracting channel flows (see Fig. 10).

5. Conclusions

A two-dimensional numerical model was implemented and applied to simulate sediment transport in the Nador lagoon.The model consists of shallow water equations for the hydrodynamics and suspended sediment coupled to bed-load trans-port for the morphodynamics. The governing equations form a hyperbolic system of conservation laws with source termsaccounting for diffusion, friction, Coriolis and erosion–deposition terms. To approximate solutions to the system we havepresented an adaptive finite volume method using unstructured triangular grids. The method consists of two stages whichcan be interpreted as a predictor-corrector procedure. In the first stage, the scheme uses the projected system of the mor-phodynamical equations and introduces the sign matrix of the flux Jacobian which results in an upwind discretization of thecharacteristic variables. In the second stage, the solution is updated using the conservative form of the equations and a spe-cial treatment of the bed bottom in order to obtain a well-balanced discretization of the flux gradient and the source term.To increase the accuracy of the scheme we have incorporated slope limiters along with an adaptive procedure using thesediment concentration as an error indicator. The considered method is second-order accurate, well balanced, nonoscillatoryand simple to implement in unstructured triangular meshes.

Verification of the proposed method has been carried out using two test cases namely, a situation with erosion–deposition effects and a case without accounting for erosion–deposition terms. The obtained results exhibited good shape,high accuracy and stability behaviour for all morphodynamical regimes considered. The presented results demonstrate thecapability of the finite volume method that can provide insight to complex sediment transport behaviours. Finally, it shouldbe remarked that it is important to perform mesh adaptation with respect to all flow variables. For example, using thebed-load and the water depth or water free-surface as monitoring function the resolution in the sediment transport may beimproved. Therefore, future work will be focused on the development of error indicators for both bed-load and hydraulicvariables. It is also worth emphasizing that, the model problem considered in the current study is highly idealized, in par-ticular, the wind effects and tidal waves as in many coastal scenarios are not accounted for. However, the results make itpromising to be applicable also to real situations where, beyond the many sources of complexity, there is a more severedemand for accuracy in predicting the morphological evolution, which must be performed for long time.

Acknowledgements

This work was partly supported by a visiting grant from the Université Paris 13 in France. The authors would like tothank anonymous referees for their careful reading of the manuscript and for their thoughtful comments, which greatlyhelped to improve the presentation.

Appendix A. Eigenvalues and sign matrix

The five eigenvalues corresponding to the projected system (10) are

λ1 = 2√−Q cos

(1

)+ 2

3uη,

λ2 = 2√−Q cos

(1(θ + 2π)

)+ 2

uη,

3 3
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Fig. 9. Adapted meshes (first column), velocity vectors (second column) and sediment concentration (third column) for sediment transport with erosioMeyer-Peter and Müller formula. From top to bottom t = 1.3, 3 and 5 hours.
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Fig. 10. Evolution in time of sediment concentration at the gauge G2 in the Nador lagoon using the Grass formula and the Meyer-Peter and Müller formula.

λ3 = 2√−Q cos

(1

3(θ + 4π)

)+ 2

3uη,

λ4 = λ5 = uη,

where

θ = cos−1(

R√−Q 3

), Q = −1

9

(u2

η + 3g(h + d)),

R = uη

54

(9g(2h − d) − 2u2

η

), d = Aξ

(3 u2

η + u2τ

).

Hence, the sign matrix in (14) is defined as

sgn[Aη(W)

] = R(W) sgn[Λ(W)

]R−1(W),

where W is the Roe’s average state given by (13), R(W) and Λ(W) are respectively, the right eigenvector and the diagonalmatrices reconstructed as

R(W) =

⎛⎜⎜⎜⎜⎜⎝1 1 1 0 −1λ1 λ2 λ3 0 −uη

uτ uτ uτ 0 −uτ + 1β

C C C h −Cα2

1c2 − 1

α22

c2 − 1α2

3c2 − 1 0 1

⎞⎟⎟⎟⎟⎟⎠ ,

Λ(W) =

⎛⎜⎜⎜⎜⎝λ1 0 0 0 00 λ2 0 0 00 0 λ3 0 00 0 0 λ4 00 0 0 0 λ5

⎞⎟⎟⎟⎟⎠ ,

R−1(W) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

γ1σ1

− α2+α3σ1

βα2α3σ1

0 c2

σ1

− γ2σ2

α1+α3σ2

− βα1α3σ2

0 − c2

σ2

γ3σ3

− α1+α2σ3

βα1α2σ3

0 c2

σ3

− Ch

0 0 1h

0

−βuτ 0 β 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,

where

γ1 = c2 − u2η + λ2λ3 − βα2α3uτ , σ1 = −α2α1 + α2α3 − α3α1 + α2

1,

γ2 = c2 − u2η + λ1λ3 − βα1α3uτ , σ2 = α2α1 + α3α2 − α3α1 − α2,

2
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1766 F. Benkhaldoun et al. / Applied Numerical Mathematics 62 (2012) 1749–1766

γ3 = c2 − u2η + λ1λ2 − βα1α2uτ , σ3 = α2α1 − α3α2 − α3α1 + α2

3,

with β = 2Aξ uτ

h, α1 = λ1 − uη , α2 = λ2 − uη , α3 = λ3 − uη , and c =

√gh is the wave speed calculated at the interface of

control volume.

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