A DECOUPLED STAGGERED SCHEME FOR THE SHALLOW WATER EQUATIONS

Abstract : We present a first order scheme based on a staggered grid for the shallow water equations with topography in two space dimensions, which enjoys several properties: positivity of the water height, preservation of constant states, and weak consistency with the equations of the problem and with the associated entropy inequality. The shallow water equations form a hyperbolic system of two conservation equations (mass and momentum) which are obtained when modelling a flow whose vertical height is considered small with respect to the plane scale. The solution of such a system may develop shocks, so that the finite volume method is usually preferred for numerical simulations. Two main approaches are found: one is the colocated approach which is usually based on some approximate Riemann solver, see e.g.  and references therein; the other one is based on a staggered arrangement of the unknowns on the grid. Indeed, staggered schemes have been used for some time in the hydraulic and ocean engineering community, see e.g. [1, 2, 12]. They have been recently analysed in the case of one space dimension [5, 8], following the works on the related barotropic Euler equations, see  and references therein. In the present work, we obtain a discrete local entropy inequality; furthermore, we extend the consistency analysis of the scheme to the case of two space dimensions, and we weaken the assumptions on the estimates , namely we no longer require a bound on the BV norm of the approximate solutions, at least for the weak formulation (the passage to the limit in the entropy still necessitates a time BV boundedness). Let Ω be an open bounded domain of R 2 and let T > 0. We consider the shallow water equations with topography over the space and time domain Ω × (0, T): ∂ t h + div(hu) = 0 in Ω × (0, T), (1a) ∂ t (hu) + div(hu ⊗ u) + ∇p + gh∇z = 0 in Ω × (0, T), (1b) p = 1 2 gh 2 in Ω × (0, T), (1c) u · n = 0 on ∂Ω × (0, T), (1d) h(x, 0) = h 0 , u(x, 0) = u 0 in Ω. (1e)
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Contributor : Raphaele Herbin <>
Submitted on : Monday, June 24, 2019 - 12:19:25 PM
Last modification on : Thursday, June 27, 2019 - 1:42:30 AM

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• HAL Id : hal-02163454, version 1
• ARXIV : 1906.11001

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Raphaèle Herbin, Jean-Claude Latché, Youssouf Nasseri, Nicolas Therme. A DECOUPLED STAGGERED SCHEME FOR THE SHALLOW WATER EQUATIONS. 2019. ⟨hal-02163454⟩

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