Fully Discrete Multi-Entropy Stability of High-Order Schemes for Compressible MHD: A Weak-to-Strong Framework

Abstract

We establish a fully discrete weak-to-strong (W2S) multi-entropy stability theory for arbitrarily high-order finite-volume and discontinuous Galerkin (DG) approximations of the ideal compressible magnetohydrodynamics (MHD) equations on general polytopal meshes, whereby a single numerical update simultaneously satisfies discrete entropy inequalities for any prescribed finite family of convex Harten entropy pairs. Because physical entropies are defined only for positive density and pressure, the central analytical difficulty lies in reconciling discrete entropy stability with positivity preservation, the magnetic divergence constraint, and the nonconservative Godunov--Powell coupling. Building upon the provably positivity-preserving MHD framework of Wu and Shu, we develop a weak multi-entropy analysis of the common cell-average evolution of finite-volume and DG methods. Through convex decomposition and relative entropy, we establish two sufficient criteria for this weak stability general polytopal meshes. Compatible magnetic corrections and locally divergence-free approximations cooperatively offset the nonconservative Godunov--Powell coupling and provide the discrete cancellations essential to both positivity-preservation and entropy stability. A W2S lifting then unifies positivity and entropy limiting into a single cellwise scaling limiter, achieving strong multi-entropy stability while strictly preserving cell averages and the locally divergence-free magnetic field. High-order temporal accuracy follows from variable-step strong-stability-preserving multistep methods. Numerical experiments confirm the theoretical findings and demonstrate computational robustness. This framework provides a rigorous, fully discrete foundation for unifying physical admissibility, magnetic divergence control, and multi-entropy stability in high-order MHD approximations.