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Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system

A lot of well-balanced schemes have been proposed for discretizing the classical Saint-Venant system for shallow water flows with non-flat bottom. Among them, the hydrostatic reconstruction scheme is a simple and efficient one. It involves the knowledge of an arbitrary solver for the homogeneous problem (for example Godunov, Roe, kinetic,...). If this solver is entropy satisfying, then the hydrostatic reconstruction scheme satisfies a semi-discrete entropy inequality. In this paper we prove that, when used with the classical kinetic solver, the hydrostatic reconstruction scheme also satisfies a fully discrete entropy inequality, but with an error term. This error term tends to zero strongly when the space step tends to zero, including solutions with shocks. We prove also that the hydrostatic reconstruction scheme does not satisfy the entropy inequality without error term.


Introduction
The classical Saint-Venant system for shallow water describes the height of water h(t, x) ≥ 0, and the water velocity u(t, x) ∈ R (x denotes a coordinate in the horizontal direction) in the direction parallel to the bottom. It assumes a slowly varying topography z(x), and reads (1.1) where g > 0 is the gravity constant. This system is completed with an entropy (energy) inequality We shall denote U = (h, hu) T and , G(U ) = h u 2 2 + gh 2 u the entropy and entropy fluxes without topography. The derivation of an efficient, robust and stable numerical scheme for the Saint-Venant system has received an extensive coverage. The issue involves the notion of well-balanced schemes, and we refer the reader to [10,17,15,19] and references therein.
The hydrostatic reconstruction (HR), introduced in [1], is a general and efficient method that uses an arbitrary solver for the homogeneous problem, like Roe, relaxation, or kinetic solvers. It leads to a consistent, well-balanced, positive scheme satisfying a semi-discrete entropy inequality, in the sense that the inequality holds only in the limit when the timestep tends to zero. The method has been generalized to balance all subsonic steady-states in [11], and to multi-layer shallow water in [12], with the source-centered variant of the hydrostatic reconstruction. The HR technique has also been used to derive efficient and robust numerical schemes approximating the incompressible Euler and Navier-Stokes equations with free surface [5,3], i.e. non necessarily shallow water flows.
The aim of this paper is to prove that the hydrostatic reconstruction, when used with the classical kinetic solver [8,4,18,9,2,16,13], satisfies a fully discrete entropy inequality. However, as established in Proposition 3.9, this inequality necessarily involves an error term. The main result of this paper is that this error term is in the square of the topography increment, ensuring that it tends to zero strongly as the space step tends to zero, for solutions that can include shocks. The topography needs however to be Lipschitz continuous.
In general, to satisfy an entropy inequality is a criterion for the stability of a scheme. In the fully discrete case, it enables in particular to get an a priori bound on the total energy. In the time-only discrete case and without topography, the single energy inequality that holds for the kinetic scheme ensures the convergence [7]. The fully discrete case (still without topography) has been treated in [6]. Another approach to get a scheme satisfying a fully discrete entropy inequality is proposed in [14].
The outline of the paper is as follows. We recall in Section 2 the kinetic scheme without topography and its entropy analysis, in both the discrete and semi-discrete cases. We show in particular how one can see that the fully discrete inequality is always less dissipative than the semi-discrete one. In Section 3 we analyze the entropy inequality of the kinetic solver with topography. We propose a kinetic interpretation of the hydrostatic reconstruction and we give its properties. The semi-discrete scheme is analyzed first. Our main result Theorem 3.7 concerning the fully discrete scheme is finally proved.
Before going into discretized models, we end this section by recalling the classical kinetic Maxwellian equilibrium, used in [18] for example, at the continuous level. The kinetic Maxwellian is given by where ξ ∈ R and x + ≡ max(0, x) for any x ∈ R. It satisfies the following moment relations, These definitions allow us to obtain a kinetic representation of the Saint-Venant system.
for some "collision term" Q(t, x, ξ) which satisfies, for a.e. (t, x), Proof. Using (1.5), the proof relies on a very obvious computation.
The interest of the particular form (1.4) lies in its link with a kinetic entropy. Consider the kinetic entropy, where f ≥ 0, ξ ∈ R and z ∈ R, and its version without topography Then one can check the relations One has the following subdifferential inequality and entropy minimization principle.
(ii) For any f (ξ) ≥ 0, setting h = f (ξ)dξ, hu = ξf (ξ)dξ (assumed finite), one has Proof. This approach by the subdifferential inequality is explained in [8]. The property (ii) obviously follows from (i) by taking f = f (ξ) and integrating (1.10) with respect to ξ. For proving (i), notice first that where prime denotes differentiation with respect to U = (h, hu) T . Thus Observe also that (1.14) where here prime denotes differentiation with respect to f . The formula defining M in (1.4) yields that We conclude using the convexity of H 0 with respect to f that which proves the claim.
We would like to approximate the solution U (t, x), x ∈ R, t ≥ 0 of the system (1.1) by discrete values U n i , i ∈ Z, n ∈ N. In order to do so, we consider a grid of points x i+1/2 , i ∈ Z, and we define the cells (or finite volumes) and their lengths We consider discrete times t n with t n+1 = t n + ∆t n , and we define the piecewise constant functions U n (x) corresponding to time t n and z(x) as ). The functions F l/r (U l , U r ) ∈ R 2 are the numerical fluxes. In the present work, the expressions for F l/r (U l , U r ) are based on the kinetic description given in Proposition 1.1.
Indeed the method used in [18] in order to solve (1.1) can be viewed as solving for the unknown f (t, x, ξ), over the time interval (t n , t n+1 ), with initial data Defining the update as the formula (1.23) can then be written This formula can in fact be written under the form (1. 19), (1.20) for some numerical fluxes F l/r computed in [18], involving nonexplicit integrals. Here we would like to use simplified formulas, and it will be done by choosing an approximation of f n+1− i (ξ). We shall often denote U i instead of U n i , whenever there is no ambiguity.

Kinetic entropy inequality without topography
In this section we consider the problem (1.1) without topography, and the unmodified kinetic scheme (1.21), (1.22), (1.24), (1.25). This problem is classical, and we recall here how the entropy inequality is analyzed in this case, in the fully discrete and semi-discrete cases.
2.1. Fully discrete scheme. Without topography, the kinetic scheme is an entropy satisfying flux vector splitting scheme [9]. The update (1.24) of the solution of (1.21),(1.22) simplifies to the discrete kinetic scheme with σ i = ∆t n /∆x i and with short notation (we omit the variable ξ). One can write it Then under the CFL condition that This can be also written as which can be interpreted as a conservative kinetic entropy inequality. Note that with (1.25) and (1.11), which by integration of (2.5) yields the macroscopic entropy inequality.
The scheme (2.1) and the definition (1.25) allow to complete the definition of the macroscopic scheme (1.19), (1.20) with the numerical flux F l = F r ≡ F given by the flux vector splitting formula [9] (2.7) where M is defined in (1.4).

2.2.
Semi-discrete scheme. Assuming that the timestep is very small (i.e. σ i very small), we have the linearized approximation of the entropy variation from (2.1) . This linearization with respect to ∆t n (or equivalently with respect to σ i = ∆t n /∆x i ) represents indeed the entropy in the semi-discrete limit ∆t n → 0 (divide (2.8) by ∆t n and let formally ∆t n → 0). The entropy inequality attached to this linearization can be estimated as follows.
Lemma 2.1. The linearized term from (2.8) is dominated by the conservative difference from (2.5), In particular, the semi-discrete scheme is more dissipative than the fully discrete scheme.
Proof. It is enough to prove two inequalities, . We observe that (2.10) is trivial for ξ > 0, and (2.11) is trivial for ξ < 0. The two conditions can therefore be written These last inequalities follow from the convexity of H 0 .

Kinetic interpretation of the hydrostatic reconstruction scheme
The hydrostatic reconstruction scheme (HR scheme for short) for the Saint-Venant system (1.1), has been introduced in [1], and can be written as follows, F is a numerical flux for the system without topography, and the reconstructed states are defined by Then, looking for a kinetic interpretation of the HR scheme, we would like to approximate the solution to (1.21) in order to write down a kinetic scheme such that the associated macroscopic scheme is exactly and we consider the scheme In this formula, δM i+1/2± depend on ξ, U i , U i+1 , ∆z i+1/2 = z i+1 − z i , and are assumed to satisfy the moment relations Using again (1.25), the integration of (3.6) multiplied by 1 ξ with respect to ξ then gives obviously the HR scheme 3.1. Analysis of the semi-discrete scheme. Assuming that the timestep is very small (i.e. σ i very small), we have the linearized approximation of the entropy variation from (3.6), (3.9) where the kinetic entropy H(f, ξ, z) is defined in (1.6). As in Subsection 2.2, this linearization with respect to σ i = ∆t n /∆x i represents indeed the entropy in the semi-discrete limit ∆t n → 0. Its dissipation can be estimated as follows.
Proposition 3.1. We assume that the extra variations δM i+1/2± satisfy (3.7), (3.8), and also Then the linearized term from (3.9) is dominated by a quasi-conservative difference, Moreover, the integral with respect to ξ of the last two lines of (3.12) (respectively of (3.13)) vanishes. In particular, Proof. The value of the integral with respect to ξ of the two last lines of (3.12) is (3.4). The computation for (3.13)) is similar. In order to prove (3.11), it is enough to prove the two inequalities We note that the definitions of h i+1/2± in (3.4)-(3.5) ensure that h i+1/2− ≤ h i , and h i+1/2+ ≤ h i+1 . Therefore, because of (1.4) one has Taking into account (3.10), with (1.16) we get The first inequality (3.23) is trivial for ξ > 0, and the second inequality (3.24) is trivial for ξ < 0. Therefore it is enough to satisfy the two inequalities (3.25) But as in Subsection 2.2, we have according to the convexity of H with respect to f , In order to prove (3.25), we observe that if M i (ξ) = 0 then M i+1/2− (ξ) = 0 also, thus (3.5), and the inequality (3.25) follows from (3.27). Next, if M i (ξ) > 0, one has and as in (1.17) Taking the difference between (3.30) and (3.29), we obtain (3.31) because of the definition (3.4) of h i+1/2− . Therefore we conclude that in any case (M i (ξ) being zero or not), one has (3.32) because of (3.27), and this proves (3.25). Similarly one gets proving (3.26). This concludes the proof, and we observe that we have indeed a dissipation estimate slightly stronger than (3.11), Remark 3.2. The numerical entropy flux (3.15) can be written where G is the numerical entropy flux of the scheme without topography, and F 0 is the first component of F . This formula is in accordance of the analysis of the semi-discrete entropy inequality in [1].
Remark 3.3. At the kinetic level, the entropy inequality (3.11) is not in conservative form. The entropy inequality becomes conservative only when taking the integral with respect to ξ, as is seen on (3.14). This is also the case in [18]. Indeed we have written the macroscopic conservative entropy inequality as an integral with respect to ξ of the sum of a nonpositive term (the one in (3.11)), a kinetic conservative term (the difference of the first lines of (3.12) and (3.13)), and a term with vanishing integral (difference of the two last lines of (3.12) and (3.13)). However, such a decomposition is not unique.

3.2.
Analysis of the fully discrete scheme. We still consider the scheme (3.6), and we make the choice  Proof. The expression kinetic well-balanced means that we do not only prove that at rest, but the stronger property when u i = 0 and h i + z i = h i+1 + z i+1 for all i. Indeed in this situation one has U i+1/2− = U i+1/2+ for all i, thus the first three terms between parentheses in (3.6) give ξM i , and the last three terms give −ξM i , leading to (3.38). The consistency of the HR scheme has been proved in [1], but here the statement is the consistency of the kinetic update (3.6) with the kinetic equation (1.21). We proceed as follows. Using (1.22) and (1.4), the topography source term in (1.21) reads This formula is valid for 2gh − (ξ − u) 2 = 0, i.e. when ξ = u ± √ 2gh or in L 1 (ξ ∈ R). Assuming that h i > 0 (otherwise the consistency is obvious), one has that h i+1/2− = h i + z i − z i+1/2 for z i+1 − z i small enough, and an asymptotic expansion of M i+1/2− gives Similarly, one has With the usual shift of index i due to the distribution of the source to interfaces, the difference (3.42) minus (3.43) appears as a discrete version of (3.39). The other four terms in parentheses in (3.6) are conservative, and are classically consistent with ξ∂ x f in (1.21).
Remark 3.5. The scheme (3.6) can be viewed as a consistent well-balanced scheme for (1.21), except that the notion of consistency is true here only for Maxwellian initial data. On the contrary, the exact solution used in [18] is consistent for initial data of arbitrary shape. The role of the special form of the Maxwellian (1.4) is seen here by the fact that for initial data U i at rest, one has that M (U i , ξ) is a steady state of (1.21) (this results from (3.39) and (3.41)).
Remark 3.6. Going one step further in the asymptotic expansion (3.40) gives (3.44) that can also be written Therefore, the relation (3.45) When writing the entropy inequality for the fully discrete scheme, the difficulty is to estimate the positive part of the entropy dissipation by something that tends to zero when ∆x i tends to zero, at constant courant number σ i , and assuming only that ∆z/∆x is bounded (Lipschitz topography), but not that ∆U/∆x is bounded (the solution can have discontinuities). Here ∆z stands for a quantity like z i+1 − z i , and ∆U stands for a quantity like U i+1 − U i .
The principle of proof of such entropy inequality is that we use the dissipation of the semi-discrete scheme proved in Proposition 3.1, under the strong form (3.34). This inequality involves the terms linear in σ i . Under a CFL condition, the higher order terms (quadratic in σ i or higher) are either treated as errors if they are of the order of ∆z 2 or ∆z∆U , or must be dominated by the dissipation if they are of the order of ∆U 2 . Note that the dissipation in (3.34), i.e. the two last expressions in factor of 1 I ξ<0 and 1 I ξ>0 respectively, are of the order of (M i+1/2+ − M i+1/2− ) 2 and (M i−1/2+ − M i−1/2− ) 2 respectively, and thus neglecting the terms in ∆z, they control (M i+1 − M i ) 2 and (M i − M i−1 ) 2 respectively. However, the Maxwellian (1.4) is not Lipschitz continuous with respect to U , thus a sharp analysis has to be performed in order to use the dissipation.
We consider a velocity v m ≥ 0 such that for all i, This means equivalently that |u i | + √ 2gh i ≤ v m . We consider a CFL condition strictly less than one, where σ i = ∆t n /∆x i , and β is a given constant.
Theorem 3.7 has the following corollary.
Corollary 3.8. Integrating the estimate (3.48) with respect to ξ, using (1.11), (1.25) and (3.14) (neglecting the dissipation) and Lemma 3.13 yields that which is the discrete entropy inequality associated to the HR scheme (3.1)-(3.5), (2.7). Note that with (3.3)-(3.5) one has We conclude that the quadratic error term (divide (3.49) by ∆t n to be consistent with (1.2)) has the following key properties: it vanishes identically when z = cst (no topography) or when σ i → 0 (semi-discrete limit), and as soon as the topography is Lipschitz continuous, it tends to zero strongly when the grid size tends to 0 (consistency with the continuous entropy inequality (1.2)), even if the solution contains shocks.
We state now a counter result saying that it is not possible to remove the error term in (3.48), even at the level of its integral with respect to ξ. It is indeed true for the HR scheme even if the homogeneous flux used is not the kinetic one.
Proposition 3.9. The HR scheme (3.1)-(3.5) does not satisfy the fully-discrete entropy inequality (3.49) without quadratic error term, whatever restrictive is the CFL condition.
Remark 3. 10. An open problem is to establish the fully discrete entropy inequality with error (3.49) for an HR scheme with general (non kinetic) homogeneous numerical flux F satisfying a fully discrete entropy inequality.
In the proofs given below, the Lemma (3.13) of L 2 −Lipschitz dependency of the Maxwellian is used. Note that the Maxwellian (1.4) is only 1/2-Hölder continuous at fixed ξ.
Proof of Theorem 3.7. Using (3.6) and (3.36), one has for ξ ≤ 0 (3.51) Then, we write the linearization of H around the Maxwellian M i is nothing but the dissipation of the semi-discrete scheme, that has been estimated in Proposition 3.1. Thus, multiplying (3.34) by −σ i , using the form (1.6) of H and the identity we get (3.55) Then, using again the form of H and (3.54), the quadratic term L i in (3.53) can be expressed as Using (3.51), we have for any α > 0 and similarly with (3.52) Therefore, adding the estimates (3.53), (3.55), (3.57), (3.58) yields (3.59) where (3.60) and α > 0 is an arbitrary parameter. Notice that the first two lines in (3.60) are basically nonpositive, whereas the third line is nonnegative.
Before going further in the proof of Theorem 3.7, i.e. upper bounding d i by a sum of a dissipation term and an error, let us state a lemma, that gives another expression for d i , in which the nonpositive contributions appear clearly.
Lemma 3.11. The term d i from (3.60) can also be written Proof of Lemma 3.11. The expression (3.51) of f n+1− i for ξ ≤ 0 allows to precise the value of d i in (3.60), and gives for ξ ≤ 0 Using (3.52) we obtain analogously for ξ ≥ 0 These expressions yield the formulas (3.61)-(3.63).
Continuation of the proof of Theorem 3.7. One would like the first two lines of (3.61) to be nonpositive. In order to get nonnegative coefficients γ But since β < 1, one can choose α > 0 such that and then the coefficients (3.65) are positive, and γ − i+1/2 , γ + i−1/2 ≥ 0. We denote Then we have Next we write using (3.51), (3.52) and (3.19) We can estimate the first quadratic error term from (3.61) as Finally we estimate and similarly (3.73) We set which is positive if ǫ is taken small enough (recall that α > 0 has been chosen so as to satisfy (3.66), and hence depends only on β). Then using (3.59) and (3.74), the two first lines in the right-hand side of (3.74) give a dissipation as stated in (3.48), while the last lines give an error. From (3.74) and (3.71), for ξ < 0 the typical error terms take the form (3.76) The term proportional to ǫ 2 can therefore be absorbed by ν 0 β . Since a similar estimate holds for ξ > 0, diminishing slightly ν 0 β by something proportional to ǫ 2 (taken small enough), we get a coefficient ν β > 0. The only remaining error terms finally take the form stated in the last line of (3.48). This completes the proof of (ii) in Theorem 3.7.
Remark 3.12. Consider the situation when for some i 0 one has with z i0−1 = z i0 or z i0 = z i0+1 . Then by (3.3), (3.4), the reconstructed states satisfy U i+1/2− = U i+1/2+ for i = i 0 − 1, i 0 . We observe that then, in the formula (3.60) for d i , the dissipative terms vanish for i = i 0 , for all ξ. Thus d i0 ≥ 0 and d i0 (ξ)dξ > 0, which means that the extra term d i in (3.59) gives a dissipation with the wrong sign, in agreement with Proposition 3.9.
Proof of Proposition 3.9. It has been proved in [1] that the semi-discrete HR scheme (limit σ i → 0) satisfies the entropy inequality without error term. Here we prove that the fully-discrete scheme does not, whatever restrictive is the CFL condition. This result holds for an arbitrary numerical flux F taken for the homogeneous Saint-Venant system. The argument is as follows.
Consider the local dissipation where U n+1 i is given by (3.1), F i+1/2± are defined by (3.2)-(3.5), and where G is the numerical entropy flux associated to F . Then, taking into account that h n+1 i = h i − σ i (F 0 (U i+1/2− , U i+1/2+ ) − F 0 (U i−1/2− , U i−1/2+ )), one has (3.79) The entropy η being strictly convex, the function is convex, and strictly convex if Assuming that this condition holds, we get that the right-hand side of (3.79) is strictly increasing with respect to σ i . In particular, it will be strictly positive if the limit as σ i → 0 of this quantity vanishes. This limit is nothing else than the dissipation of the semi-discrete scheme Consider data such that (3.83) U i = U l , z i = z l for i ≤ i 0 , U i = U r , z i = z r for i > i 0 , for left and right states U l = (h l , h l u l ), U r = (h r , h r u r ) such that (3.84) u l = u r = 0, h l + z l = h r + z r , z r − z l > 0.
Then one checks easily that (3.81) holds for i = i 0 , and that (3.82) vanishes for all i. Therefore, D n i0 > 0, which proves the claim. Lemma 3.13. Let U k = (h k , h k u k ) for k = 1, 2, 3 with h k ≥ 0. Then We can also estimate M (U 1 , ξ) by M (U 1 , ξ) + 2M (U 2 , ξ), giving the same estimate as (3.87) with U 1 and U 2 exchanged and with an extra factor 2. This proves (3.85). Then, denoting M k ≡ M (U k , ξ), according to the Minkowsky inequality,