Non-uniform convergence of finite volume schemes for Riemann problems of ideal magnetohydrodynamics

Abstract

This paper presents Riemann test problems for ideal magnetohydrodynamics (MHD) finite volume schemes. The test problems place emphasis on the hyperbolic irregularities of the ideal MHD system, namely the occurrence of intermediate shocks and non-unique solutions. We investigate numerical solutions for the test problems obtained by several commonly used methods (Roe, HLLE, central scheme). All methods turned out to show a non-uniform convergence behavior which may be paraphrased as ‘pseudo-convergence’: Initially the methods show convergence towards a wrong solution with irregular wave patterns. Only after heavy grid refinement the methods switch to converge to the true solution. This behavior is most pronounced whenever the solution should be unique but its phase space trajectory lies in the vicinity of that of a non-unique solution. We show detailed grid convergence studies and empirical error analysis. The results may be related to similar results for a 2 × 2 model system and to time dependent investigations of other authors. The non-uniform convergence is expected to be present for any diffusive finite volume method and question the reliability of coarse grid ideal MHD solutions, also in higher space dimensions.