Multigrid Methods for Systems of Hyperbolic Conservation Laws

Abstract

In this paper, we present total variation diminishing (TVD) multigrid methods for computing the steady state solutions for systems of hyperbolic conservation laws. An efficient multigrid method should avoid spurious numerical oscillations. This can be achieved by designing methods which preserve monotonicity and TVD properties through the use of upwind interpolation and restriction techniques. Such multigrid methods have been developed for scalar conservation laws in the literature. However, for hyperbolic systems, the upwinding directions are not apparent. We generalize the upwind techniques to systems by formulating the interpolation as solving a local Riemann problem. Upwind biased restriction is performed on the positive and negative components of the residual. This idea stems from the fact that the flux vector can be split into positive and negative components. For two-dimensional systems, we introduce a novel coarsening technique and extend the upwind interpolation and restriction techniques, which together capture the characteristics of the underlying system of hyperbolic equations. We provide a theoretical analysis to show that our two level method is TVD for one-dimensional linear systems. We also prove that both the additive and multiplicative multigrid schemes are consistent and convergent for one-dimensional linear systems. We demonstrate the effectiveness of our method by numerical examples including Euler equations.