A Provably Positive Discontinuous Galerkin Method for Multidimensional Ideal Magnetohydrodynamics
A Provably Positive Discontinuous Galerkin Method for Multidimensional Ideal Magnetohydrodynamics
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DOI：
10.1137/18m1168042
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发表时间：
2018-07
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影响因子：
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通讯作者：
Kailiang Wu;Chi-Wang Shu
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文献类型：
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作者：
Kailiang Wu;Chi-Wang Shu
在磁流体力学（MHD）中，密度和压力是正物理量。理想可压缩MHD的可证明保正（PP）数值格式的设计是非常可取的，但仍然是一个挑战，特别是在多维情况下。本文给出了一致高阶不连续伽辽金（DG）格式，该格式可证明保持了多维理想MHD的密度和压力的正性。采用可对称理想MHD方程的局部无散度DG格式作为基格式，利用PP限制器增强DG解的正性，并采用时间离散化的强稳定性保持方法构造了该格式。重要的创新是，我们使用一种新的可容许状态集的等价形式和非常技术性的估计，发现并严格证明了所提出的DG方案的PP性质。几个二维数值算例进一步证实了PP的性质，并证明了该方法的准确性。
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 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, ...