Provably positivity-preserving, globally divergence-free central DG methods for ideal MHD system

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.

The authors' abstract, as published at the source. Mathematical Models and Methods in Applied Sciences, 2026 · DOI ↗