Free-Stream Preserving Finite Difference Schemes for Ideal Magnetohydrodynamics on Curvilinear Meshes

Abstract

In this paper, a high order free-stream preserving finite difference weighted essentially non-oscillatory (WENO) scheme is developed for the ideal magnetohydrodynamic (MHD) equations on curvilinear meshes. Under the constrained transport framework, magnetic potential evolved by a Hamilton–Jacobi (H–J) equation is introduced to control the divergence error. In this work, we use the alternative formulation of WENO scheme (Christlieb et al. in SIAM J Sci Comput 40(4):A2631–A2666, 2018) for the nonlinear hyperbolic conservation law, and design a novel method to solve the magnetic potential. Theoretical derivation and numerical results show that the scheme can preserve free-stream solutions of MHD equations, and reduce error more effectively than the standard finite difference WENO schemes for such problems.