论文标题
通过几何形状的准线性化，可证明的正中央DG方案可用于理想的MHD方程
Provably Positive Central DG Schemes via Geometric Quasilinearization for Ideal MHD Equations
论文作者
论文摘要
在理想MHD的数值模拟中，保持压力和密度为阳性对于物理考虑和数值稳定性至关重要。这是一个挑战，这是由于这种具有阳性性（PP）特性与无磁发射（DF）约束以及MHD方程的强非线性之间的基本关系。本文介绍了中央不连续的Galerkin（CDG）方法的第一个严格的PP分析，并构建了理想MHD的任意高阶PP CDG方案。通过最近开发的几何学学（GQL）方法，我们的分析表明，标准CDG方法的PP属性与离散的DF条件密切相关，其形式未知，与非中心DG和有限的体积案例不同[K。吴，暹罗J.肛门。 2018]。该结果为我们的PP CDG方案的设计奠定了基础。在1D情况下，自然满足离散的DF条件，我们证明标准CDG方法在可以使用PP限制器执行的条件下是PP。但是，在多维情况下，离散的DF条件是高度不平凡但至关重要的，并且我们证明，即使使用PP限制仪，我们也证明了标准的CDG方法通常不是PP，因为它无法满足离散的DF条件。我们通过仔细分析离散差异的结构，然后为Godunov修改后的MHD方程构建新的本地DF CDG方案来解决这个问题。关键点是找出源术语的合适离散化，以便它可以完全抵消离散DF条件下的所有术语。基于GQL方法，我们证明了新的多维CDG方案的PP属性。 PP CDG方案的鲁棒性和准确性通过几个苛刻的例子进行了验证，包括高速射流和血浆非常低的爆炸问题。
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.
