Provably Positive Central DG Schemes via Geometric Quasilinearization for Ideal MHD Equations

Abstract

In the numerical simulation of ideal magnetohydrodynamics (MHD), keeping the pressure and density always positive is essential for both physical considerations and numerical stability. This is however a challenging task, due to the underlying relation between such positivity-preserving (PP) property and the magnetic divergence-free (DF) constraint as well as the strong nonlinearity of the MHD equations. In this paper, we present the first rigorous PP analysis of the central discontinuous Galerkin (CDG) methods and construct arbitrarily high-order provably PP CDG schemes for ideal MHD. By the recently developed geometric quasilinearization (GQL) approach, our analysis reveals that the PP property of standard CDG methods is closely related to a discrete magnetic DF condition, whose form was yet unknown prior to our analysis and differs from that for the non-central DG and finite volume methods in [K. Wu, SIAM J. Numer. Anal., 56 (2018), pp. 2124-2147. The discovery of this relation lays the foundation for the design of our PP CDG schemes. In the 1D case, the discrete DF condition is naturally satisfied, and we rigorously prove that the standard CDG method is PP under a condition that can be enforced easily with an existing PP limiter. However, in the multidimensional cases, the corresponding discrete DF condition is highly nontrivial yet critical, and we analytically prove that the standard CDG method, even with the PP limiter, is not PP in general, as it generally fails to meet the discrete DF condition. We address this issue by carefully analyzing the structure of the discrete divergence terms and then constructing new locally DF CDG schemes for Godunov's modified MHD equations with an additional source term. The key point is to find out the suitable discretization of the source term such that it exactly cancels out all the terms in the discovered discrete DF condition. Based on the GQL approach, we prove in theory the PP property of the new multidimensional CDG schemes under a CFL condition. The robustness and accuracy of the proposed PP CDG schemes are further validated by several demanding numerical MHD examples, including the high-speed jets and blast problems with very low plasma beta.