Efficient Admissible Set Projection in Optimization-based Invariant-Domain-Preserving Limiters for Ideal MHD

TL;DR

This paper introduces an efficient optimization-based limiter for ideal MHD equations that enforces admissibility, preserves conservation, and enhances robustness in high-order DG schemes through a novel one-dimensional projection approach.

Contribution

The authors develop a decomposition-based projection method that simplifies the admissibility enforcement in MHD to a one-dimensional problem, improving efficiency and robustness.

Findings

The projection reduces to a one-dimensional minimization solvable by Brent's method.

The limiter enforces cell average and pointwise admissibility in DG schemes.

Numerical tests demonstrate improved robustness and physical fidelity in MHD simulations.

Abstract

Preserving the admissible set of the ideal magnetohydrodynamics (MHD) equations is important not only for producing physically meaningful numerical solutions, but more importantly for achieving robust computations. In this paper, we develop an optimization-based limiter to enforce admissibility while preserving global conservation and accuracy. For an easy and efficient projection, we decompose the admissible set into slices parameterized by the magnetic energy, so that the MHD projection reduces to a one-dimensional minimization, which can be solved efficiently by the Brent method. The splitting method can be used to efficiently solve the global minimization problem of the optimization-based limiter, which can be used to enforce cell average admissibility in discontinuous Galerkin (DG) schemes, and pointwise admissibility can be further enforced by the Zhang-Shu positivity-preserving limiter. We apply the limiter to high-order DG schemes and present numerical results for a few representative MHD problems.