An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment

Bouchut, François; Tomás Morales de Luna

ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 4, pp. 683-698.

We consider the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water. The difficulty in this system comes from the coupling terms involving some derivatives of the unknowns that make the system nonconservative, and eventually nonhyperbolic. Due to these terms, a numerical scheme obtained by performing an arbitrary scheme to each layer, and using time-splitting or other similar techniques leads to instabilities in general. Here we use entropy inequalities in order to control the stability. We introduce a stable well-balanced time-splitting scheme for the two-layer shallow water system that satisfies a fully discrete entropy inequality. In contrast with Roe type solvers, it does not need the computation of eigenvalues, which is not simple for the two-layer shallow water system. The solver has the property to keep the water heights nonnegative, and to be able to treat vanishing values.

[1] R. Abgrall and S. Karni, Computations of compressible multifluids. J. Comput. Phys. 169 (2001) 594-623. | MR | Zbl

[2] R. Abgrall and S. Karni, A relaxation scheme for the two-layer shallow water system, in Proceedings of the 11th International Conference on Hyperbolic Problems (Lyon, 2006), Hyperbolic problems: theory, numerics, applications, S. Benzoni-Gavage and D. Serre Eds., Springer (2007) 135-144.

[3] E. Audusse and M.-O. Bristeau, A well-balanced positivity preserving “second-order” scheme for shallow water flows on unstructured meshes. J. Comput. Phys. 206 (2005) 311-333. | MR | Zbl

[4] E. Audusse, M.-O. Bristeau and B. Perthame, Kinetic schemes for Saint-Venant equations with source terms on unstructured grids. INRIA Report, RR-3989 (2000).

[5] E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (2004) 2050-2065 (electronic). | MR | Zbl

[6] D.S. Bale, R.J. Leveque, S. Mitran and J.A. Rossmanith, A wave propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM J. Sci. Comput. 24 (2002) 955-978 (electronic) | MR | Zbl

[7] M. Baudin, C. Berthon, F. Coquel, R. Masson and Q.H. Tran, A relaxation method for two-phase flow models with hydrodynamic closure law. Numer. Math. 99 (2005) 411-440. | MR

[8] C. Berthon and F. Coquel, Travelling wave solutions of a convective diffusive system with first and second order terms in nonconservation form, in Hyperbolic problems: theory, numerics, applications, Vol. I (Zürich, 1998), Internat. Ser. Numer. Math. 129, Birkhäuser, Basel (1999) 74-54. | MR | Zbl

[9] F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Frontiers in Mathematics. Birkhäuser Verlag, Basel (2004). | MR | Zbl

[10] F. Bouchut, S. Medvedev, G. Reznik, A. Stegner and V. Zeitlin, Nonlinear dynamics of rotating shallow water: methods and advances, Edited Series on Advances in Nonlinear Science and Complexity. Elsevier (2007).

[11] M. Castro, J. Macías and C. Parés, A $Q$-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system. ESAIM: M2AN 35 (2001) 107-127. | Numdam | MR | Zbl

[12] Q. Jiang and R.B. Smith, Ideal shocks in a 2-layer flow. II: Under a passive layer. Tellus 53A (2001) 146-167.

[13] J.B. Klemp, R. Rotunno and W.C. Skamarock, On the propagation of internal bores. J. Fluid Mech. 331 (1997) 81-106. | Zbl

[14] M. Li and P.F. Cummins, A note on hydraulic theory of internal bores. Dyn. Atm. Oceans 28 (1998) 1-7.

[15] C. Parés and M. Castro, On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems. ESAIM: M2AN 38 (2004) 821-852. | Numdam | MR | Zbl

[16] M. Pelanti, F. Bouchut, A. Mangeney and J.-P. Vilotte, Numerical modeling of two-phase gravitational granular flows with bottom topography, in Proc. of HYP06, Lyon, France (2007).

[17] B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201-231. | MR | Zbl

[18] J.B. Schijf and J.C. Schonfeld, Theoretical considerations on the motion of salt and fresh water, in Proc. of the Minn. Int. Hydraulics Conv., Joint meeting IAHR and Hyd. Div. ASCE (1953) 321-333.