ADER scheme on unstructured meshes for shallow water: Simulation of tsunami waves

In this paper, we extend Arbitrary accuracy DErivatives Riemann (ADER)-type finite volume numerical methods to simulate tsunami wave propagation and steady state flows with geometrical sources. The mathematical formulation of the physical problem is based on the shallow water equations with non-constant bathymetry. Although, the physical process described by the shallow water equations preserves the steady state condition balancing fluxes and sources, a number of finite volumes numerical methods fail to preserve this 'well-balanced' property at the discrete level. In the last few years many numerical methods have been presented that fulfil this condition although retaining high-order accuracy. However, most of them are based on a 1-D quadrature rule and are suitable only for Cartesian grids. Here we present an arbitrary high-order finite volume scheme based on the ADER approach on 2-D unstructured meshes that preserves steady state solutions with arbitrarily high order of accuracy. This property allows us to properly represent tsunami wave propagation over realistic bathymetries.