可压缩磁流体动力学高阶格式的完全离散多熵稳定性：一个弱到强框架

中文摘要 AI 辅助

我们针对一般多面体网格上理想可压缩磁流体动力学（MHD）方程的任意高阶有限体积和间断伽辽金（DG）近似，建立了完全离散的弱到强（W2S）多熵稳定性理论，通过该理论，单次数值更新可同时满足任意给定的凸Harten熵对有限族的离散熵不等式。由于物理熵仅对正密度和正压力定义，核心分析难点在于协调离散熵稳定性与保正性、磁场散度约束以及非守恒的Godunov-Powell耦合。基于Wu和Shu的可证明保正的MHD框架，我们发展了有限体积和DG方法公共单元平均值演化的弱多熵分析。通过凸分解和相对熵，我们建立了该弱稳定性在一般多面体网格上的两个充分判据。相容的磁场修正和局部无散近似协同抵消非守恒的Godunov-Powell耦合，并提供对保正性和熵稳定性都至关重要的离散抵消。随后，W2S提升将保正性和熵限制统一为单个单元级缩放限制器，在严格保持单元平均值和局部无散磁场的同时实现强多熵稳定性。高阶时间精度由变步长强稳定性保持多步方法保证。数值实验证实了理论发现并展示了计算鲁棒性。该框架为在高阶MHD近似中统一物理可容许性、磁场散度控制和多熵稳定性提供了严格的完全离散基础。

英文摘要

We establish a fully discrete weak-to-strong (W2S) multi-entropy stability theory for arbitrarily high-order finite-volume and discontinuous Galerkin (DG) approximations of the ideal compressible magnetohydrodynamics (MHD) equations on general polytopal meshes, whereby a single numerical update simultaneously satisfies discrete entropy inequalities for any prescribed finite family of convex Harten entropy pairs. Because physical entropies are defined only for positive density and pressure, the central analytical difficulty lies in reconciling discrete entropy stability with positivity preservation, the magnetic divergence constraint, and the nonconservative Godunov--Powell coupling. Building upon the provably positivity-preserving MHD framework of Wu and Shu, we develop a weak multi-entropy analysis of the common cell-average evolution of finite-volume and DG methods. Through convex decomposition and relative entropy, we establish two sufficient criteria for this weak stability general polytopal meshes. Compatible magnetic corrections and locally divergence-free approximations cooperatively offset the nonconservative Godunov--Powell coupling and provide the discrete cancellations essential to both positivity-preservation and entropy stability. A W2S lifting then unifies positivity and entropy limiting into a single cellwise scaling limiter, achieving strong multi-entropy stability while strictly preserving cell averages and the locally divergence-free magnetic field. High-order temporal accuracy follows from variable-step strong-stability-preserving multistep methods. Numerical experiments confirm the theoretical findings and demonstrate computational robustness. This framework provides a rigorous, fully discrete foundation for unifying physical admissibility, magnetic divergence control, and multi-entropy stability in high-order MHD approximations.

发表机构

• Southern University of Science and Technology（南方科技大学）
• Shenzhen International Center for Mathematics, Southern University of Science and Technology（南方科技大学深圳数学中心）

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