一种用于复杂几何域上波动方程的能量型间断伽辽金方法
An energy-based discontinuous Galerkin method for wave equations on complex geometric domains
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中文总结 AI 辅助
本文提出一种高阶切割能量型间断伽辽金方法，用于复杂域及界面上的声波方程，通过ghost-penalty稳定化解决小切割单元问题，保持高阶精度和稳定能量，数值实验验证了收敛性和鲁棒性。
中文摘要 AI 辅助
本文针对复杂域及跨越固定材料界面上的二阶声波方程，提出了一种高阶切割能量型间断伽辽金（CutEDG）方法。该方法采用非贴体笛卡尔网格，并通过将ghost-penalty稳定化直接引入定义EDG能量的局部双线性形式中，解决了小切割单元问题，同时保持原有的空间耦合和数值通量结构不变。我们建立了半离散能量稳定性以及先验误差估计，表明高阶精度得以保持。该公式进一步扩展到具有不连续波速的界面问题。数值实验证实了预测的收敛阶数，并展示了对于任意小的切割单元的鲁棒性。特别地，显式时间步长限制由背景网格尺寸而非最小物理切割单元决定，且稳定化的能量Gram矩阵表现出与切割无关的条件数。
英文摘要
This paper develops a high-order cut energy-based discontinuous Galerkin (CutEDG) method for second-order acoustic wave equations on complex domains and across stationary material interfaces. The method employs unfitted Carte- sian meshes and addresses the small-cut-cell problem by introducing ghost-penalty stabilization directly into the local bilinear forms defining the EDG energy, while leaving the original spatial coupling and numerical-flux structure unchanged. We establish semidiscrete energy stability and a priori error estimates showing that high-order accuracy is retained. The formulation is further extended to interface problems with discontinuous wave speeds. Numerical experiments confirm the pre- dicted convergence rates and demonstrate robustness with respect to arbitrarily small cut cells. In particular, the explicit time-step restriction is governed by the background mesh size rather than the smallest physical cut cell, and the stabilized energy Gram matrices exhibit cut-independent conditioning.
