高阶间断伽辽金方法的一种子单元细化熵残差驱动限制策略
A subcell-refined entropy-residual-driven limiting strategy for high-order discontinuous Galerkin methods
- College of Mathematics and System Sciences, Xinjiang University（新疆大学数学与系统科学学院）
- School of Aeronautics and Astronautics, Shanghai Jiao Tong University（上海交通大学航空学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对高阶DG方法，提出子单元细化熵残差驱动限制策略，以最小耗散恢复熵不等式，保持精度并严格熵耗散，显著简化保正过程。
AI中文摘要：
在欠分辨状态下，细粒度的子单元级耗散控制对于实现非线性双曲系统的高阶间断伽辽金（DG）模拟的鲁棒性同时保持精度至关重要。本文针对勒让德-高斯-洛巴托节点上的DG方法提出了一种子单元细化熵残差驱动的限制策略。该限制器仅在每个单元内引入最近邻成对耗散，其闭式系数提供恢复单元熵不等式所需的最小耗散。该策略是经典熵稳定方法的对角、局部稳定近似，一个广义子单元框架揭示了分裂形式DG和基于残差分布的熵修正方案作为限制系数的特定选择。对于欧拉方程，一个物理一致的跳跃算子分别模拟热和剪切熵产生，同时保持速度和压力平衡；对Zhang-Shu保正限制器的子单元细化确保逐点正性。大量数值测试证实，该格式保持最优高阶精度，严格强制熵耗散，并显著降低了后验保正过程的难度。
英文摘要：
Fine-grained, subcell-level dissipation control is essential for achieving robust high-order discontinuous Galerkin (DG) simulations of nonlinear hyperbolic systems in under-resolved regimes while preserving accuracy. This paper proposes a subcell-refined entropy-residual-driven limiting strategy for DG on Legendre-Gauss-Lobatto nodes. The limiter introduces only nearest-neighbor pairwise dissipation within each element, with closed-form coefficients that supply the minimal dissipation required to restore the element entropy inequality. The strategy is a diagonal, locally stable approximation of classical entropy-stable methods, and a generalized subcell framework reveals split-form DG and residual-distribution-based entropy correction schemes as particular choices of the limiting coefficients. For the Euler equations, a physically consistent jump operator separately models thermal and shear entropy production while preserving velocity and pressure equilibrium; a subcell refinement of the Zhang-Shu positivity limiter ensures pointwise positivity. Extensive numerical tests confirm that the scheme maintains optimal high-order accuracy, strictly enforces entropy dissipation, and significantly reduces the difficulty of a posteriori positivity-preserving procedures.
