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. Well-balancing 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.

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