We present an assessment of the accuracy of a recently developed MHD code used to study hydromagnetic flows in supernovae and related events. The code, based on the constrained transport formulation, incorporates unprecedented ultra-high-order methods (up to 9th order) for the reconstruction and the most accurate approximate Riemann solvers. We estimate the numerical resistivity of these schemes in tearing instability simulations.

We study a depth-averaged model of gravity-driven flows made ofsolid grains and fluid, moving over variable basal surface. In particular, we are interested in applicationsto 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 thesolid 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 leastwhen 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 oftopography source terms is obtained via a technique that includesthese contributions into the wave structure of the Riemann solution. We present and discuss several numerical experiments, including problemsof perturbed steady flows over non-flat bottom surface that show the efficient modeling of disturbances of equilibrium conditions.