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

Abstract

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. For multidimensional conservative MHD system, standard DG methods with a PP limiter are not PP in general, due to the effect of unavoidable divergence-error. We construct provably PP high-order DG and finite volume schemes by proper discretization of symmetrizable MHD system, with two divergence-controlling techniques: locally divergence-free elements and a penalty term. The former leads to zero divergence within each cell, while the latter controls the divergence error across cell interfaces. Our analysis reveals that a coupling of them is important for positivity preservation, as they exactly contribute the discrete divergence-terms absent in standard DG schemes but crucial for ensuring the PP property. Numerical tests confirm the PP property and the effectiveness of proposed PP schemes. Unlike conservative MHD system, the exact smooth solutions of symmetrizable MHD system are proved to retain the positivity even if the divergence-free condition is not satisfied. Our analysis and findings further the understanding, at both discrete and continuous levels, of the relation between the PP property and the divergence-free constraint.

49 pages, 11 figures

References in corpus (16)

• An Unsplit Godunov Method for Ideal MHD via Constrained Transport
• Second Order Accurate Schemes for Magnetohydrodynamics With Divergence-Free Reconstruction
• Positivity-preserving flux limiters for high-order conservative schemes
• A Divergence-Free Upwind Code for Multidimensional Magnetohydrodynamic Flows
• Multidimensional HLLE Riemann solver; Application to Euler and Magnetohydrodynamic Flows
• A robust numerical scheme for highly compressible magnetohydrodynamics: Nonlinear stability, implementation and tests
• High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics
• Ideal GLM-MHD: About the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations
• Finite Difference Weighted Essentially Non-Oscillatory Schemes with Constrained Transport for Ideal Magnetohydrodynamics
• Provably Positive Discontinuous Galerkin Methods for Multidimensional Ideal Magnetohydrodynamics
• Positivity-Preserving Analysis of Numerical Schemes for Ideal Magnetohydrodynamics
• Design of Provably Physical-Constraint-Preserving Methods for General Relativistic Hydrodynamics
• Admissible state and physical constraints preserving schemes for relativistic magnetohydrodynamic equations
• A high-order positivity-preserving single-stage single-step method for the ideal magnetohydrodynamic equations
• Physical-constraints-preserving central discontinuous Galerkin methods for special relativistic hydrodynamics with a general equation of state
• Runge-Kutta discontinuous Galerkin methods for the special relativistic magnetohydrodynamics

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