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 [C. R. Acad. Sci. Paris, Serie I, 323, 77-84, 1996]. The relaxation
interpretation allows establishing positivity conditions for this VFRoe method.
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