Provably Positive Central DG Schemes via Geometric Quasilinearization for Ideal MHD Equations
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In the numerical simulation of ideal MHD, keeping the pressure and density positive is essential for both physical considerations and numerical stability. This is a challenge, 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. This paper presents the first rigorous PP analysis of the central discontinuous Galerkin (CDG) methods and constructs arbitrarily high-order 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 DF condition, whose form was unknown and differs from the non-central DG and finite volume cases in [K. Wu, SIAM J. Numer. Anal. 2018]. This result lays the foundation for the design of our PP CDG schemes. In 1D case, the discrete DF condition is naturally satisfied, and we prove the standard CDG method is PP under a condition that can be enforced with a PP limiter. However, in the multidimensional cases, the discrete DF condition is highly nontrivial yet critical, and we prove the the standard CDG method, even with the PP limiter, is not PP in general, as it fails to meet the discrete DF condition. We address this issue by carefully analyzing the structure of the discrete divergence and then constructing new locally DF CDG schemes for Godunov's modified MHD equations with an additional source. The key point is to find out the suitable discretization of the source term such that it exactly offsets all the terms in the discrete DF condition. Based on the GQL approach, we prove the PP property of the new multidimensional CDG schemes. The robustness and accuracy of PP CDG schemes are validated by several demanding examples, including the high-speed jets and blast problems with very low plasma beta.
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