A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers
- Departement de Mathematiques et Applications, Ecole Normale Superieure, 45, rue d'Ulm - 75230 Paris cedex 05 (France)
- CNRS and Departement de Mathematiques et Applications, Ecole Normale Superieure, 45, rue d'Ulm - 75230 Paris cedex 05 (France)
- Equipe de Sismologie, Institut de Physique du Globe de Paris, 4, place Jussieu - 75252 Paris cedex 05 (France)
We present a Riemann solver derived by a relaxation technique for classical single-phase shallow flow equations and for a two-phase shallow flow model describing a mixture of solid granular material and fluid. Our primary interest is the numerical approximation of this two-phase solid/fluid model, whose complexity poses numerical difficulties that cannot be efficiently addressed by existing solvers. In particular, we are concerned with ensuring a robust treatment of dry bed states. The relaxation system used by the proposed solver is formulated by introducing auxiliary variables that replace the momenta in the spatial gradients of the original model systems. The resulting relaxation solver is related to Roe solver in that its Riemann solution for the flow height and relaxation variables is formally computed as Roe's Riemann solution. The relaxation solver has the advantage of a certain degree of freedom in the specification of the wave structure through the choice of the relaxation parameters. This flexibility can be exploited to handle robustly vacuum states, which is a well known difficulty of standard Roe's method, while maintaining Roe's low diffusivity. For the single-phase model positivity of flow height is rigorously preserved. For the two-phase model positivity of volume fractions in general is not ensured, and a suitable restriction on the CFL number might be needed. Nonetheless, numerical experiments suggest that the proposed two-phase flow solver efficiently models wet/dry fronts and vacuum formation for a large range of flow conditions. As a corollary of our study, we show that for single-phase shallow flow equations the relaxation solver is formally equivalent to the VFRoe solver with conservative variables of Gallouet and Masella [T. Gallouet, J.-M. Masella, Un schema de Godunov approche C.R. Acad. Sci. Paris, Serie I, 323 (1996) 77-84]. The relaxation interpretation allows establishing positivity conditions for this VFRoe method.
- OSTI ID:
- 21499760
- Journal Information:
- Journal of Computational Physics, Vol. 230, Issue 3; Other Information: DOI: 10.1016/j.jcp.2010.10.001; PII: S0021-9991(10)00545-0; Copyright (c) 2010 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; ISSN 0021-9991
- Country of Publication:
- United States
- Language:
- English
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