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

TL;DR

This paper introduces PosDiv-CDG, a high-order numerical method for ideal MHD that guarantees positivity and divergence-free magnetic fields, with rigorous proofs and robustness demonstrated in extreme plasma conditions.

Contribution

It develops a provably positivity-preserving, divergence-free central DG method for MHD, resolving structural conflicts in existing schemes with novel strategies and rigorous analysis.

Findings

Proves positivity preservation under explicit CFL conditions.

Demonstrates robustness in high Mach number MHD jet simulations.

Maintains high-order accuracy and non-oscillatory behavior near shocks.

Abstract

This paper proposes a numerical method, termed PosDiv-CDG, that provably preserves both positivity and the globally divergence-free (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 central discontinuous Galerkin (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 cell 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.