Finite volume schemes for multi-dimensional hyperbolic systems based on the use of bicharacteristics

Abstract

In this paper we present recent results for the bicharacteristic based finite volume schemes, the so-called finite volume evolution Galerkin (FVEG) schemes. These methods were proposed to solve multi-dimensional hyperbolic conservation laws. They combine the usually conflicting design objectives of using the conservation form and following the characteristics, or bicharacteristics. This is realized by combining the finite volume formulation with approximate evolution operators, which use bicharacteristics of multi-dimensional hyperbolic system. In this way all of the infinitely many directions of wave propagation are taken into account. The main goal of this paper is to present a self contained overview on the recent results. We study the L1-stability of the finite volume schemes obtained by different approximations of the flux integrals. Several numerical experiments presented in the last section confirm robustness and correct multi-dimensional behaviour of the FVEG methods.