A Riemann Solver for Single-Phase and Two-Phase Shallow Flow Models based on Relaxation. Relations with Roe and VFRoe Solvers.
Résumé
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 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 is exploited for a robust handling of wet/dry fronts and vacuum formation. Several numerical experiments, including tests of dry bed generation, show the effectiveness of the approach. The proposed relaxation solver is related to Roe's solver in that its Riemann solution for the flow height and relaxation variables is formally computed as Roe's Riemann solution. Furthermore, for single-phase shallow flow equations, the relaxation solver is found to be formally equivalent to the VFRoe solver with conservative variables of Gallouët and Masella [C.R. Acad. Sc. Paris, Série I, 323, 77--84, 1996]. The relaxation interpretation allows to establish positivity conditions for this VFRoe method.
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