Abstract.We study a depth-averaged model of gravity-driven flows made of solid grains and fluid, moving over variable basal surface. In particular, we are interested in applications to geophysical flows such as avalanches and debris flows, which typically contain both solid material and interstitial fluid. The model system consists of mass and momentum balance equations for the solid and fluid components, coupled together by both conservative and non-conservative terms involving the derivatives of the unknowns, and by interphase drag source terms. The system is hyperbolic at least when the difference between solid and fluid velocities is sufficiently small. We solve numerically the one-dimensional model equations by a high-resolution finite volume scheme based on a Roe-type Riemann solver. Wellbalancing of topography source terms is obtained via a technique that includes these contributions into the wave structure of the Riemann solution. We present and discuss several numerical experiments, including problems of perturbed steady flows over non-flat bottom surface that show the efficient modeling of disturbances of equilibrium conditions. Mathematics Subject Classification. 65M99, 76T25.

International audienceWe model liquid-gas flows with cavitation by a variant of the six-equation single-velocity two-phase model with stiff mechanical relaxation of Saurel–Petitpas–Berry [J. Comput. Phys. 228 (2009), 1678–1712]. In our approach we employ phasic total energy equations instead of the phasic internal energy equations of the classical six-equation system. This alternative formulation allows us to easily design a simple numerical method that ensures consistency with mixture total energy conservation at the discrete level and agreement of the relaxed pressure at equilibrium with the correct mixture equation of state. Temperature and Gibbs free energy exchange terms are included in the equations as relaxation terms to model heat and mass transfer and hence liquid-vapor transition. The algorithm uses a high-resolution wave propagation method for the numerical approximation of the homogeneous hyperbolic portion of the model. In two dimensions a fully-discretized scheme based on a hybrid HLLC/Roe Riemann solver is employed. Thermo-chemical terms are handled numerically via a stiff relaxation solver that forces thermodynamic equilibrium at liquid-vapor interfaces under metastable conditions. We present numerical results of sample tests in one and two space dimensions that show the ability of the proposed model to describe cavitation mechanisms and evaporation wave dynamics

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International audienceWe show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48, 235 (1995)] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of approximate Riemann solvers is proposed which allows as many as 2m waves in the resulting solution. These solvers are related to more general relaxation systems and connections with several other standard solvers are explored. The added flexibility of 2m waves may be advantageous in deriving new methods. Some potential applications are explored for problems with discontinuous flux functions or source terms

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