Development and application of generalized MUSTA schemes

Abstract

Numerical methods for solving non-linear systems of hyperbolic conservation laws via finite volume methods or discontinuous Galerkin finite element methods require, as the building block, a monotone numerical flux. The simplest approach for providing a monotone numerical utilizes a symmetric stencil and does not explicitly make use of wave propagation information, giving rise to centred or symmetric schemes. A more refined approach utilizes wave propagation information through the exact or approximate solution of the Riemann problem, giving rise to Godunov methods. Conventional approximate Riemann solvers are usually complex and are not available for many systems of practical interest, such as for models for compressible multi-phase flows. It is thus desirable to construct a numerical flux that emulates the best flux available (upwind) with the simplicity and generality of symmetric schemes. Here we build upon MUSTA approach [2,3], which leads to schemes that have the simplicity and generality of symmetric schemes and the accuracy of upwind schemes. First we present a new flux that is an average of symmetric fluxes and which reproduces the Godunov upwind scheme for the model hyperbolic equation. For non-linear systems it is found that this flux gives superior results to those of the whole family of incomplete Riemann solvers that do not explicitly account for linearly degenerate fields. Then we incorporate this flux into the MUSTA multi-staging approach, as predictor and corrector. It is found that the resulting MUSTA schemes reproduce the Godunov upwind scheme for the model hyperbolic equation for any number of stages, including multiple space dimensions. They are linearly stable in two and three space dimensions and the stability region is identical to that of the Godunov upwind method. For non-linear systems the MUSTA scheme with one or two stages gives results that are indistinguishable from those of Riemann solvers, such as the exact Riemann solver or Roe approximate Riemann solver. Finally, we assess the schemes on carefully chosen test problems for the one-dimensional equations of magneto hydrodynamics and nonlinear elasticity. High-order examples are provided for the multidimensional Euler equations in the framework of finite-volume WENO schemes [1]. The results illustrate the accuracy and efficiency of new methods combined with the ease of coding.