Numerical path preserving Roe scheme for ideal MHD Riemann problem: Complete elimination of pseudo-convergence

Abstract

Pseudo-convergence appears when ideal magnetohydrodynamic (MHD) equations are numerically solved, i.e., a converged numerical solution cannot be obtained even by continuously refining the grids under the initial condition of a large angle of the tangential magnetic field. However, the current numerical methods for pseudo-convergence have not explored the intrinsic cause of pseudo-convergence. Therefore, the current numerical schemes cannot completely eliminate the pseudo-convergence phenomenon. In this paper, we first perform an in-depth analysis of pseudo-convergence and find that the essence of pseudo-convergence lies in the unphysical averaging process of the existing numerical schemes for the Alfvénic waves. Based on this finding, the merits of numerical path preserving (NPP) of the Godunov scheme are generalized to correct the eigenvalues, eigenvectors, and wave strength of the Alfvénic field in the MHD Roe scheme, and the novel NPP-Roe scheme is constructed so that the Alfvénic field can be captured correctly. Compared with the traditional Roe scheme, numerical validation shows that NPP-Roe scheme significantly reduces the computational grid requirements for the numerical simulation of the MHD problem and eliminates the pseudo-convergence phenomenon of the MHD problem by directly reducing the absolute error magnitude. In addition, for Riemann problems with tangential symmetry (e.g., 180° Alfvénic wave, Brio & Wu problem), the NPP-Roe scheme is also able to simulate the exact regular solutions that cannot be obtained by the traditional Roe scheme, which indicates that the NPP-Roe scheme expands the application range of the traditional scheme.