An upwind differencing scheme for the equations of ideal magnetohydrodynamics
Journal of Computational Physics
Non-oscillatory central differencing for hyperbolic conservation laws
Journal of Computational Physics
On Godunov-type methods near low densities
Journal of Computational Physics
A numerical study of a rotationally degenerate hyperbolic system. Part I: The Riemann problem
Journal of Computational Physics
A higher-order Godunov method for multidimensional ideal magnetohydrodynamics
SIAM Journal on Scientific Computing
An approximate Riemann solver for ideal magnetohydrodynamics
Journal of Computational Physics
Extension of the piecewise parabolic method to multidimensional ideal magnetohydrodynamics
Journal of Computational Physics
Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves
SIAM Journal on Mathematical Analysis
A simple Riemann solver and high-order Godunov schemes for hyperbolic systems of conservation laws
Journal of Computational Physics
Shock-capturing approach and nonevolutionary solutions in magnetohydrodynamics
Journal of Computational Physics
Journal of Computational Physics
A High-Order Iterative Implicit-Explicit Hybrid Scheme for Magnetohydrodynamics
SIAM Journal on Scientific Computing
A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics
Journal of Computational Physics
An entropic solver for ideal lagrangian magnetohydrodynamics
Journal of Computational Physics
A solution-adaptive upwind scheme for ideal magnetohydrodynamics
Journal of Computational Physics
A high-order gas-kinetic method for multidimensional ideal magnetohydrodynamics
Journal of Computational Physics
High order WENO schemes: investigations on non-uniform converges for MHD Riemann problems
Journal of Computational Physics
Journal of Computational Physics
Linear high-resolution schemes for hyperbolic conservation laws: TVB numerical evidence
Journal of Computational Physics
Compact third-order limiter functions for finite volume methods
Journal of Computational Physics
Hi-index | 31.47 |
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.