基于网格的滤波方法以缓解球极坐标下Runge–Kutta间断Galerkin方法的时间步长限制：在欧拉方程中的应用
Mesh-Based Filtering to Alleviate Time-Step Restrictions in Runge--Kutta Discontinuous Galerkin Methods in Spherical-polar Coordinates: Application to the Euler Equations
AI总结：
该研究提出一种基于网格的滤波方法，缓解球极坐标下RKDG方法的时间步长限制，通过辅助合并网格消除单元各向异性，将其应用于欧拉方程，大幅提升模拟的稳定时间步长以加速计算。
AI中文摘要：
我们提出了一种基于网格的滤波方法，用于缓解球极坐标下显式Runge–Kutta间断Galerkin（RKDG）方法中出现的严重时间步长限制。该滤波方法可在原始逻辑笛卡尔网格上实现稳定演化，同时使用与辅助合并网格相关联的更大时间步长；该辅助合并网格用于消除坐标奇点附近汇聚的坐标线所产生的极端单元各向异性。该滤波方法被实现为s级RK时间积分器内的一系列后处理操作，因此可直接集成到现有结构化网格DG框架中。我们在一维空间维度中对该滤波方法进行了分析，并证明滤波后的RKDG方法等价于在通过合并基础均匀网格的选定单元得到的非均匀网格上进行RKDG离散化演化。该等价性意味着滤波后的方法继承了合并网格上对应RKDG离散化的精度与稳定性特性。我们将该基于网格的滤波方法应用于球极坐标下欧拉方程的现有RKDG方法，并通过选定的二维和三维算例，证明其通过显著更大的稳定时间步长来加速模拟的有效性。
英文摘要：
We propose a mesh-based filtering approach to alleviate the severe timestep restrictions arising in explicit Runge--Kutta discontinuous Galerkin (RKDG) methods formulated in spherical-polar coordinates. The filter enables stable evolution on the original logically Cartesian mesh while using larger time steps associated with an auxiliary merged mesh constructed to eliminate the extreme cell anisotropies produced by converging coordinate lines near coordinate singularities. The filter is implemented as a sequence of post-processing operations applied within an $s$-stage RK time integrator, making it straightforward to incorporate into existing structured-mesh DG frameworks. We analyze the filter in one spatial dimension and prove that the filtered RKDG method is equivalent to evolving the RKDG discretization on a nonuniform mesh obtained by merging selected elements of the underlying uniform mesh. This equivalence implies that the filtered method inherits the accuracy and stability properties of the corresponding RKDG discretization on the merged mesh. We apply the mesh-based filter to an existing RKDG method for the Euler equations in spherical-polar coordinates and demonstrate, through selected two- and three-dimensional examples, its effectiveness in accelerating simulations through significantly larger stable timesteps.
