A wave propagation method for three-dimensional hyperbolic conservation laws

Abstract

A class of wave propagation algorithms for three-dimensional conservation laws and other hyperbolic systems is developed. These unsplit finite-volume methods are based on solving one-dimensional Riemann problems at the cell interfaces and applying flux-limiter functions to suppress oscillations arising from second-derivative terms. Waves emanating from the Riemann problem are further split by solving Riemann problems in the transverse directions to model cross-derivative terms. With proper upwinding, a method that is stable for Courant numbers up to 1 can be developed. The stability theory for three-dimensional algorithms is found to be more subtle than in two dimensions and is studied in detail. In particular we find that some methods which are unconditionally unstable when no limiter is applied are (apparently) stabilized by the limiter function and produce good looking results. Several computations using the Euler equations are presented including blast wave and complex shock/vorticity problems. These algorithms are implemented in the software, which is freely available.