Finite Volume Evolution Galerkin Methods for Hyperbolic Systems

Abstract

The subject of the paper is the derivation and analysis of new multidimensional, high-resolution, finite volume evolution Galerkin (FVEG) schemes for systems of nonlinear hyperbolic conservation laws. Our approach couples a finite volume formulation with approximate evolution operators. The latter are constructed using the bicharacteristics of the multidimensional hyperbolic system such that all of the infinitely many directions of wave propagation are taken into account. In particular, we propose a new FVEG scheme, which is designed in such a way that for a linear wave equation system the approximate evolution operator calculates any one-dimensional planar wave exactly. This operator improves the stability of the FVEG scheme considerably, leading to a stability limit closer to 1. Using the results obtained for the wave equation system, a new approximate evolution operator for the linearized Euler equations is also derived. The integrals over the cell interfaces also need to be approximated with care; in this case, our choice of Simpson's rule is guided by stability analysis of model problems. Second order resolution is obtained by means of a piecewisebilinear recovery. Numerical experiments confirm the accuracy and multidimensional behavior of the new scheme.