Provably Positive High-Order Schemes for Ideal Magnetohydrodynamics: Analysis on General Meshes

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This paper proposes and analyzes arbitrarily high-order discontinuous Galerkin (DG) and finite volume methods which provably preserve the positivity of density and pressure for the ideal MHD on general meshes.

• Unified auxiliary theories are built for rigorously analyzing the positivity-preserving (PP) property of MHD schemes with a HLL type flux on polytopal meshes in any space dimension.
• The main challenges overcome here include establishing relation between the PP property and discrete divergence of magnetic field on general meshes, and estimating proper wave speeds in the HLL flux to ensure the PP property.
• In 1D case, we prove that the standard DG and finite volume methods with the proposed HLL flux are PP, under condition accessible by a PP limiter.