Provably Positive Discontinuous Galerkin Methods for Multidimensional Ideal Magnetohydrodynamics
This provides a rigorous solution to a long-standing challenge in computational MHD, enabling stable simulations of low-density or low-pressure regions.
The authors developed the first provably positivity-preserving high-order discontinuous Galerkin schemes for multidimensional ideal magnetohydrodynamics, ensuring density and pressure remain positive. Numerical tests in 2D confirm accuracy and robustness.
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.