This article presents high order accurate discontinuous Galerkin (DG) methods for wave problems on moving curved meshes with general choices of basis and quadrature. The proposed method adopts an arbitrary Lagrangian–Eulerian formulation to map the wave equation from a time-dependent moving physical domain onto a fixed reference domain. For moving curved meshes, weighted mass matrices must be assembled and inverted at each time step when using explicit time-stepping methods. We avoid this step by utilizing an easily invertible weight-adjusted approximation. The resulting semi-discrete weight-adjusted DG scheme is provably energy stable up to a term that (for a fixed time interval) converges to zero with the same rate as the optimal $L^2$ error estimate. Numerical experiments using both polynomial and B-spline bases verify the high order accuracy and energy stability of proposed methods.

中文翻译：

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用于移动曲线网格上波传播的高阶权重调整不连续伽辽金方法

本文针对移动曲线网格上的波浪问题提出了高阶精确的不连续伽辽金 (DG) 方法，具有一般的基和正交选择。所提出的方法采用任意的拉格朗日-欧拉公式将波动方程从与时间相关的移动物理域映射到固定参考域。对于移动曲线网格，当使用显式时间步长方法时，必须在每个时间步长组装和反转加权质量矩阵。我们通过使用一个容易可逆的权重调整近似来避免这一步。由此产生的半离散权重调整 DG 方案可证明是能量稳定的，直到（对于固定的时间间隔）以与最佳的速率相同的速率收敛到零${升}^2$误差估计。使用多项式和 B 样条基的数值实验验证了所提出方法的高阶精度和能量稳定性。