Provably Positive Discontinuous Galerkin Methods for Multidimensional Ideal Magnetohydrodynamics

TL;DR

This paper introduces high-order discontinuous Galerkin schemes for multidimensional ideal MHD that provably preserve positivity of density and pressure, ensuring physically meaningful simulations with demonstrated accuracy and robustness.

Contribution

The paper develops and rigorously proves positivity-preserving high-order DG schemes for multidimensional ideal MHD, a significant advancement in numerical MHD methods.

Findings

Schemes maintain positivity of density and pressure in multidimensional MHD.

Numerical examples confirm accuracy and robustness of the methods.

Theoretical proof of the PP property using novel admissible state set analysis.

Abstract

The density and pressure are positive physical quantities in magnetohydrodynamics (MHD). Design of provably positivity-preserving (PP) numerical schemes for ideal compressible MHD is highly desirable, but remains a challenge especially in the multidimensional cases. In this paper, we first develop uniformly high-order discontinuous Galerkin (DG) schemes which provably preserve the positivity of density and pressure for multidimensional ideal MHD. The schemes are constructed by using the locally divergence-free DG schemes for the symmetrizable ideal MHD equations as the base schemes, a PP limiter to enforce the positivity of the DG solutions, and the strong stability preserving methods for time discretization. The significant innovation is that we discover and rigorously prove the PP property of the proposed DG schemes by using a novel equivalent form of the admissible state set and very technical estimates. Several two-dimensional numerical examples further confirm the PP property, and demonstrate the accuracy, effectiveness and robustness of the proposed PP methods.