Abstract
The compressible MHD equations possess two essential structural properties: (i) an algebraic structure ensuring the positivity of density and pressure; and (ii) a differential structure maintaining the divergence-free (DF) constraint on the magnetic field. A deep connection between these two properties has been revealed in both non-central [K. Wu, SIAM J. Numer. Anal., 56 (2018), pp. 2124–2147] and central discontinuous Galerkin (CDG) frameworks [K. Wu, H. Jiang & C.-W. Shu, SIAM J. Numer. Anal., 61 (2023), pp. 250–285]. However, existing methods could provably preserve positivity only in conjunction with a locally DF property. Constructing a uniformly high-order method that is both provably positive and globally DF has remained an open and challenging problem.
This paper proposes a numerical method, termed PosDiv-CDG, that provably preserves both positivity and the global DF condition at arbitrarily high order in multiple dimensions. It resolves the fundamental structural incompatibility between standard positivity-preserving limiters and global DF enforcement in the CDG framework. The method integrates a novel positivity-limiting strategy, a modified dissipation mechanism guided by convex decomposition, and an auxiliary evolution equation for the magnetic field, which are designed based on rigorous theoretical analysis. Notably, we provide a rigorous proof of positivity preservation for the updated auxiliary full-state averages under an explicit CFL-type condition. The proof leverages the geometric quasi-linearization (GQL) technique, which reformulates the nonlinear positivity constraint into an equivalent linear form. This enables the derivation of flux-based inequalities and technical estimates under the global DF constraint. To suppress nonphysical oscillations near shocks, we develop a compact, non-intrusive convex-oscillation-suppressing (COS) procedure based on the entropy function. The COS process acts only on non-magnetic variables, avoids costly characteristic decomposition, and maintains both the globally DF property and high-order accuracy. Several challenging experiments—including low plasma-beta MHD jets with Mach numbers up to 1,000,000—demonstrate the proposed method robustness, high-order accuracy, non-oscillatory behavior, and its ability to preserve both positivity and globally DF structures under extreme conditions.