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A simple fully well-balanced and entropy preserving

scheme for the shallow-water equations

Christophe Berthon, Victor Michel-Dansac

To cite this version:

Christophe Berthon, Victor Michel-Dansac. A simple fully well-balanced and entropy preserving

scheme for the shallow-water equations. Applied Mathematics Letters, Elsevier, 2018, 86, pp.284-290.

�10.1016/j.aml.2018.07.013�. �hal-01708991v2�

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A simple fully well-balanced and entropy preserving scheme for the

shallow-water equations

Christophe Berthona, Victor Michel-Dansacb,∗

aLaboratoire de Mathématiques Jean Leray, Université de Nantes, 2 rue de la Houssinière, 44322 Nantes Cedex 3, France bInstitut de Mathématiques de Toulouse, INSA Toulouse et Université Toulouse 3 Paul Sabatier, 135 avenue de Rangueil,

31077 Toulouse Cedex 4, France

Abstract

In this communication, we consider a numerical scheme for the shallow-water system. The scheme under consideration has been proven to preserve the positivity of the water height and to be fully well-balanced, i.e. to exactly preserve the smooth moving steady state solutions of the shallow-water equations with the topography source term. The goal of this work is to prove a discrete entropy inequality satised by this scheme.

Keywords: shallow-water equations, Godunov-type schemes, well-balanced schemes, moving steady states, entropy-satisfying schemes

2000 MSC: 65M08, 65M12

1. Introduction

In this work, we consider the shallow-water approximation of free-surface ows in a longitudinal channel. In one space dimension, this model is governed by the following system:

     ∂th + ∂xq = 0, ∂tq + ∂x  q2 h + 1 2gh 2  = −gh∂xZ, (1) where h(x, t) is the water height, q(x, t) is its discharge (equal to hu, where u is the water velocity), g is the gravity constant and Z is the smooth given bottom topography. In order to shorten the notations, we rewrite (1) under the classical form of a conservation law with a source term ∂tW + ∂xF (W ) = S(W ), where

we have set: W =h q  , F (W ) =   q q2 h + 1 2gh 2  , S(W ) =  0 −gh∂xZ  .

In the present work, we assume that the ow is always far from dry areas. The conserved variables W thus lie in the set of admissible states Ω, which prescribes a physically admissible positive water height:

Ω = {W =t(h, q) ∈ R2| h > 0}.

In addition, note that the following natural entropy inequality, satised by the admissible entropy weak solutions, arises from this system:

∂tη(W ) + ∂xG(W ) ≤ −gq∂xZ, (2)

Corresponding author

Email addresses: christophe.berthon@univ-nantes.fr (Christophe Berthon), vmd@math.univ-toulouse.fr (Victor Michel-Dansac )

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where the entropy η and the entropy ux G are dened by: η(W ) = 1 2 q2 h + 1 2gh 2 and G(W ) = q h  1 2 q2 h + gh 2  .

Note that, according to [5], the non-conservative entropy inequality (2) can be recast under the following equivalent conservative form:

∂t(η(W ) + ghZ) + ∂x(G(W ) + gqZ) ≤ 0. (3)

Preserving the steady state solutions of the shallow-water equations, i.e. solutions of (1) such that ∂tW = 0, has been a major challenge of the last two decades. The steady states at rest, describing motionless

water over a possibly complex bottom topography, are obtained by assuming q = 0 as well as ∂tW = 0, to

get ∂x(h + Z) = 0. These steady states have been the focus of much work, and several relevant numerical

schemes have been developed (see for instance [1, 2, 4], but this list is far from being exhaustive). Such steady solutions are widely encountered in real-world applications, and being able to preserve them is crucial for a numerical method. Conversely, less work has been undertaken on so-called fully well-balanced schemes, which exactly preserve the smooth moving steady state solutions, dened by:

   q =cst, q2 2h2 + g (h + Z) =cst. (4) In particular, few rst-order schemes have been developed (see for instance [8,3,10]). Nice properties for a scheme to possess are, in addition to being well-balanced, the preservation of the admissible set Ω (i.e. the preservation of the water height positivity, also called the robustness property) and a discrete analogue to the entropy inequality (2).

In [3], the authors derive a scheme with the three previous properties. However, in practice, this scheme is computationally too costly, since it involves nding the roots of a fth-order polynomial. The scheme proposed in [10] corrects this cost shortcoming by introducing an approach taking into account a generic source term, leading to a suitable linearization. However, an entropy inequality was not exhibited. The goal of the present work is to establish an entropy inequality satised, in some sense to be prescribed, by the numerical scheme proposed in [10].

2. Presentation of the numerical scheme

In this Section, for the sake of completeness, we give the numerical scheme developed in [10]. It falls within the framework of nite volume schemes, and more specically of Godunov-type schemes (see [9] for instance). As usual, we introduce a discretization of the one-dimensional space domain R by dening cells of constant volume ∆x. The cell ci = (xi−1/2, xi+1/2) has center located at xi, with xi±1/2 = xi± ∆x/2.

The time step is denoted by ∆t and it is restricted according to the following CFL-like condition: ∆t ≤ ∆x

2Λ, with Λ = maxi∈Z

 −λL i+1/2, λ R i+1/2  , (5) where λL i+1/2and λ R

i+1/2 are approximations of the characteristic velocities u±

ghof the hyperbolic system, to be dened. As prescribed by the nite volume framework, the solution of the shallow-water system (1) is approximated, at time tn, by the following function, piecewise constant in each cell c

i:

W∆(x, tn) = Win if x ∈ (xi−1/2, xi+1/2).

Godunov's scheme is based on the exact solution of the Riemann problems arising between the piecewise constant approximations at the interfaces of two consecutive cells. Using exact solutions, however, is inadvis-able in practice. Indeed, source terms and nonlinearities prevent the derivation of analytical exact solutions

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to Riemann problems. Thus, we turn to a Godunov-type scheme, where a consistent approximation of the exact solution are introduced. The updated solution, at time tn+ ∆tand within the cell c

i, is then given by: Win+1:= 1 ∆x Z xi+1/2 xi−1/2 f W (x, tn+ ∆t) dx, (6)

where we have set f W (x, tn+ t) = fWR  x − xi+1/2 t ; W n i , W n i+1  for x ∈ (xi, xi+1),

withWfR(x/t; WL, WR) a relevant approximation of the solution to the Riemann problem for (1) between

arbitrary states WL∈ Ωand WR∈ Ω. As a consequence, the updated approximate solution at time tn+1is

also piecewise constant on each cell, and it is given by

W∆(x, tn+1) = Win+1if x ∈ (xi−1/2, xi+1/2).

In [10], the authors prescribe the following two-state approximate Riemann solver in order to recover necessary properties of consistency and well-balance:

f WR x t; WL, WR  =          WL if x/t < λL(WL, WR), WL∗(WL, WR) if λL(WL, WR) < x/t < 0, WR∗(WL, WR) if 0 < x/t < λR(WL, WR), WR if x/t > λR(WL, WR). (7) Note the presence of a stationary wave, of velocity 0, separating the states W∗

L and WR∗. This wave

corre-sponds to the action of the source term. We then get the following relation by computing the integral in (6): Win+1= Win− ∆t ∆x h λLi+1/2Wi+1/2L,∗ − Win  − λRi−1/2Wi−1/2R,∗ − Win i , (8)

where we have set:

λLi+1/2 = λL(Win, W n i+1), λ R i−1/2= λR(Wi−1n , W n i ),

Wi+1/2L,∗ = WL∗(Win, Wi+1n ), Wi−1/2R,∗ = WR∗(Wi−1n , Win).

In (7), the velocities λL(WL, WR)and λR(WL, WR) are approximations of the characteristic velocities

u ±√ghof the hyperbolic system. For instance, these approximations can be dened as follows: λL = min  −|uL| − p ghL, −|uR| − p ghR, −ελ  , λR= max  |uL| + p ghL, |uR| + p ghR, ελ  , where we set ελ = 10−10 to ensure that λL < 0 < λR. Let us underline that ner choices, where

λL 6= −λR, can be adopted (for instance see [12]). The intermediate states are WL∗(WL, WR) =t(h∗L, q∗)

and W∗

R(WL, WR) =t(h∗R, q∗), where the intermediate heights and discharge are given by:

q∗= qHLL+ S ∆x λR− λL , h∗L= hHLL− λRS ∆x α(λR− λL) , h∗R= hHLL− λLS ∆x α(λR− λL) . (9)

In (9), we have introduced the intermediate state of the HLL solver (see [9]), dened as follows: WHLL= hHLL qHLL  = λR λR− λL WR− λL λR− λL WL− 1 λR− λL (F (WR) − F (WL)) . (10)

Moreover, the quantity α is given by α = −(q∗)2/(h

LhR) + g(hL+ hR)/2. Finally, the quantity S is a

consistent approximation of the source term −gh∂xZ, given by:

S∆x = −g 2hLhR hL+ hR (ZR− ZL) + g 2 [h]3c hL+ hR , (11)

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where [h]c is a cuto of [h] = hR− hL, dened with a positive constant C that does not depend on ∆x:

[h]c=

(

hR− hL if |hR− hL| ≤ C ∆x,

sign(hR− hL) C ∆x otherwise.

After [10], this cuto turns out to be essential to ensure the required consistency of S with the continuous source term.

The numerical scheme from [10] is thus recalled. Let us nish by remarking that the time update (8) can be rewritten under the usual ux-source formulation:

Win+1= Win− ∆t ∆x  Fi+1/2n − Fi−1/2n +∆t 2  Si+1/2n + Si−1/2n , (12) where the expressions of the numerical ux function F and the numerical source term S are explicitly given in [10]. Note that the numerical scheme is in conservative form if the topography is at, i.e. if ∂xZ = 0.

We nally state the following result, proven in [10]. Theorem 2.1 ([10]). Assume that Wn

i ∈ Ωfor all i ∈ Z. Then, under the CFL-like condition (5), and for

a small enough ∆x, the scheme (12) is:

• consistent with the shallow-water system (1); • robust: for all i ∈ Z, if Wn

i ∈ Ω, then W n+1 i ∈ Ω;

• fully well-balanced: if the ow is a steady state according to (4), i.e. if there exists two constants q0

and Φ0such that, for all i ∈ Z, qin= q0and (qn

i)2

2(hn i)2

+g (hn

i + Zi) = Φ0, then, for all i ∈ Z, Win+1= Win.

3. An entropy inequality

According to [9], a Godunov-type scheme applied to the homogeneous system (i.e. with ∂xZ = 0) is

entropy-satisfying if the following inequality is satised under the CFL condition (5): 1 ∆x Z ∆x/2 −∆x/2 ηWfR  x ∆t; WL, WR  dx ≤ 1 2(η(WL) + η(WR)) − ∆t ∆x(G(WR) − G(WL)), (13) where fWR is the approximate Riemann solution (7). As soon as the topography is non-at, the above

formula contains a new term, denoted by T (WL, WR)and consistent with the source term −gq∂xZ in (2).

In addition, let us underline that, in order to correctly prove the well-known Lax-Wendro Theorem (see our main result, Theorem3.3), the estimation (13) must be divided by ∆x. As a consequence, we immediately note that (13) can relax up to O(∆x1+δ), with δ > 0 (see for instance [6, 3]). In the present work, we will

obtain a relaxed estimation with δ = 1. Also, according to the two-state denition ofWfR, the integral of the

left-hand side of (13) can be computed explicitly and recast with respect to η∗

L := η(WL∗)and η∗R:= η(WR∗),

as follows:

λRη∗R− λLη∗L≤ λRηR− λLηL− (GR− GL) + T (WL, WR) ∆x + O(∆x2), (14)

where, with clear notations, we have set ηL:= η(WL), GL:= G(WL), and so on. After [9], let us underline

the following inequality:

(λR− λL)ηHLL≤ λRηR− λLηL− (GR− GL),

where we have set ηHLL = η(WHLL), with WHLL dened by (10). Therefore, (14) holds as soon as the

following estimation is established:

λRηR∗ − λLηL∗ = (λR− λL)ηHLL+ T (WL, WR) ∆x + O(∆x2).

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Lemma 3.1. For a smooth topography function, the following estimation holds:

λRηR∗ − λLηL∗ = (λR− λL)ηHLL+ T (WL, WR) ∆x + O(∆x2), (15)

where the term T (WL, WR)is consistent with the topography source term −gq∂xZ, and it is given by:

T (WL, WR) = −g 2hLhR hL+ hR qHLL hHLL ZR− ZL ∆x . Proof: By denition (2) of the entropy function η, we immediately have

λRη∗R− λLη∗L= 1 2  λR (q∗)2 h∗ R − λL (q∗)2 h∗ L  +g 2 λR(h ∗ R) 2− λ L(h∗L) 2 . (16)

The core of this proof lies in a relevant expansion with respect to ∆x of the intermediate states h∗

L, h∗Rand

q∗given by (9). Indeed, since the topography function is assumed to be smooth, we get ZR− ZL= O(∆x).

As a consequence, the denition (11) of the approximate source term yields the following relation: S∆x = −g 2hLhR

hL+ hR

(ZR− ZL) + O(∆x3). (17)

Equipped with such an expansion, the intermediate states turn out to be a perturbation of the HLL intermediate state (10). Indeed, plugging (17) into (9), we have the following estimations:

q∗= qHLL− g˜h∆Z λR− λL +O(∆x3), h∗L= hHLL+ g˜h∆Z α λR λR− λL +O(∆x3), h∗R= hHLL+ g˜h∆Z α λL λR− λL +O(∆x3), where we have set ∆Z = ZR− ZL and ˜h = h2hLL+hhRR. As a consequence, we obtain the following relations:

(q∗)2= qHLL2 − 2qHLL g˜h∆Z λR− λL + O(∆x2), 1 h∗ L = 1 hHLL − 1 h2 HLL g˜h∆Z α λR λR− λL + O(∆x2), (h∗L)2= h2HLL+ 2hHLL g˜h∆Z α λR λR− λL + O(∆x2), 1 h∗R = 1 hHLL − 1 h2 HLL g˜h∆Z α λL λR− λL + O(∆x2), (h∗R)2= h2HLL+ 2hHLL g˜h∆Z α λL λR− λL + O(∆x2). Arguing the above relations and performing straightforward computations yields the following estima-tions: λR (q∗)2 h∗R − λL (q∗)2 h∗L = (λR− λL) qHLL2 hHLL − 2qHLL hHLL g˜h∆Z + O(∆x2), λR(h∗R)2− λL(hL∗)2= (λR− λL)h2HLL+ O(∆x 2 ). Combining these estimations into (16), we immediately get:

λRη∗R− λLη∗L= (λR− λL)  1 2 q2 HLL hHLL +g 2h 2 HLL  − g˜hqHLL hHLL ∆Z + O(∆x2),

which is nothing but the expected estimation (15). The proof is thus achieved.  We now state the discrete entropy inequality satised by the numerical scheme.

Theorem 3.2. The numerical scheme (12) satises the following discrete entropy inequality: ηin+1≤ ηn i − ∆t ∆x G(W n i , W n i+1) − G(W n i−1, W n i ) + ∆t T n i−1/2+ O(∆x 2),

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where the numerical entropy ux G is given by: G(WL, WR) = GL+ ∆x 2∆tηL− 1 ∆t Z ∆x/2 0 ηWf  x ∆t; WL, WR  dx, and where the numerical entropy source term Tn

i−1/2 is dened by T n

i−1/2 = T (W n

i−1, Win), with the

func-tion T (WL, WR)introduced in Lemma 3.1.

Proof: The proof of this result follows immediately from the application of Jensen's inequality and from Lemma3.1. For more details, the reader is referred to [11].  We conclude this work by an extension of classical Lax-Wendro theorem, whose proof can be found in [7] for instance.

Theorem 3.3 (Lax-Wendro). Let us introduce the notation W∆(x, t) = Wn

i for x ∈ (xi−1/2, xi+1/2)

and t ∈ (tn, tn+1). According to (12), there exists a consistent numerical ux F and a consistent

approxi-mation S of the topography source term −gh∂xZ such that

Win+1= Win− ∆t ∆x  Fn i+1/2− F n i−1/2  +∆t 2  Sn i+1/2+ S n i−1/2  .

In addition, according to Theorem3.2, there exists a consistent numerical entropy ux G and a consistent approximation T of the entropy topography source term −gq∂xZ such that

ηn+1i ≤ ηni −

∆t ∆x



Gi+1/2n − Gi−1/2n + ∆t Ti−1/2n + O(∆x2).

Assume that ∆x tends to 0 while preserving a constant ratio ∆t/∆x. If, in addition, W∆is valued in some

compact set K ∈ Ω, and the sequence W∆ converges in L1

loc(R × R+; Ω) towards some W , then W is an

entropy weak solution of (1).

Note that the converged solution W will immediately satisfy the non-conservative version (2) of the en-tropy inequality. However, according to [5], the conservative version (3) is equivalent to the non-conservative one, and thus the converged solution will also satisfy this conservative entropy inequality. The numerical scheme under consideration therefore converges towards an entropy weak solution of (1).

Acknowledgments. C. Berthon acknowledges the nancial support of the ANR-14-CE25-0001 ACHYLLES. V. Michel-Dansac acknowledges the nancial support of the Service d'Hydrographie et d'Océanographie de la Marine (SHOM).

[1] E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame. A fast and stable well-balanced scheme with hydro-static reconstruction for shallow water ows. SIAM J. Sci. Comput., 25(6):20502065, 2004.

[2] E. Audusse, C. Chalons and P. Ung. A simple well-balanced and positive numerical scheme for the shallow-water system. Commun. Math. Sci., 13(5):13171332, 2015.

[3] C. Berthon and C. Chalons. A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations. Math. Comp., 85(299):12811307, 2016.

[4] C. Berthon and F. Foucher. Ecient well-balanced hydrostatic upwind schemes for shallow-water equations. J. Comput. Phys., 231(15):49935015, 2012.

[5] F. Bouchut. Nonlinear stability of nite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2004.

[6] G. Gallice. Solveurs simples positifs et entropiques pour les systèmes hyperboliques avec terme source. C. R. Math. Acad. Sci. Paris, 334(8):713716, 2002.

[7] E. Godlewski and P.-A. Raviart. Numerical approximation of hyperbolic systems of conservation laws, volume 118 of Applied Mathematical Sciences. Springer-Verlag, New York, 1996.

[8] L. Gosse. A well-balanced ux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comput. Math. Appl., 39(9-10):135159, 2000.

[9] A. Harten, P. D. Lax, and B. van Leer. On upstream dierencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev., 25(1):3561, 1983.

[10] V. Michel-Dansac, C. Berthon, S. Clain, and F. Foucher. A well-balanced scheme for the shallow-water equations with topography. Comput. Math. Appl., 72(3):568593, 2016.

[11] E. F. Toro. Riemann solvers and numerical methods for uid dynamics. A practical introduction. Springer-Verlag, Berlin, third edition, 2009.

[12] J.-P. Vila. Simplied Godunov Schemes for 2 × 2 Systems of Conservation Laws. SIAM J. Numer. Anal., 1986, 23, 11731192

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