Cut meshes are a type of mesh that is formed by allowing embedded boundaries to "cut" a simple underlying mesh resulting in a hybrid mesh of cut and standard elements. While cut meshes can allow complex boundaries to be represented well regardless of the mesh resolution, their arbitrarily shaped and sized cut elements can present issues such as the small cell problem, where small cut elements can result in a severely restricted CFL condition. State redistribution, a technique developed by Berger and Giuliani [1], can be used to address the small cell problem. In this work, we pair state redistribution with a high-order discontinuous Galerkin scheme that is $L_2$ energy stable for arbitrary quadrature. We prove that state redistribution can be added to a provably $L_2$ energy stable discontinuous Galerkin method on a cut mesh without damaging the scheme's $L_2$ stability. We numerically verify the high order accuracy and stability of our scheme on two-dimensional wave propagation problems.

翻译：切割网格是一种通过允许嵌入边界“切割”简单底层网格而形成的混合网格，由切割单元和标准单元共同组成。尽管切割网格能够在任意网格分辨率下良好地表示复杂边界，但其形状与尺寸不规则的切割单元会引发诸如小单元问题——即微小切割单元可能导致CFL条件严重受限。由Berger和Giuliani [1]提出的状态再分配技术可用于解决小单元问题。本研究将状态再分配与一种对任意求积具有$L_2$能量稳定的高阶间断伽辽金格式相结合。我们证明了在切割网格上，状态再分配可被添加至一个经过严格证明的$L_2$能量稳定间断伽辽金方法中，且不会破坏该格式的$L_2$稳定性。通过二维波传播问题数值验证了该格式的高阶精度与稳定性。