Efficient Admissible Set Projection in Optimization-based Invariant-Domain-Preserving Limiters for Ideal MHD
For computational MHD, this provides an efficient method to ensure physically meaningful and robust numerical solutions.
The paper develops an optimization-based limiter for ideal MHD that enforces admissibility while preserving conservation and accuracy, using a slice decomposition to reduce projection to 1D minimization solved by Brent's method. Numerical results for high-order DG schemes demonstrate effectiveness.
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.