High-order entropy stable summation-by-parts (SBP) schemes are a class of robust and accurate numerical methods for hyperbolic conservation laws that are numerically stable at arbitrary order without the need for artificial stabilization. While SBP schemes are well-established on simplicial and tensor-product elements, they have not been extended to cut meshes. Cut meshes provide a convenient and efficient means of mesh generation for domains with embedded boundaries but can be difficult to use due to their arbitrarily shaped cut elements. Using the skew-hybridized SBP formulation of Chan ["Skew-symmetric entropy stable...", JSC, 2019], we present a high-order accurate, entropy stable scheme for hyperbolic conservation laws on cut meshes. The formulation requires positive/non-negative weight quadrature rules on cut elements, which we construct via explicit parameterizations, subtriangulations, and Caratheodory pruning. We numerically verify the accuracy and stability of our method using the shallow water and compressible Euler equations and note promising results for the use of state redistribution with entropy stable methods.

翻译：高阶熵稳定求和-分部（SBP）格式是一类用于双曲守恒律的鲁棒且精确的数值方法，其在任意阶数下均具有数值稳定性，无需人工稳定化。虽然SBP格式在单纯形和张量积单元上已得到充分确立，但尚未扩展到切割网格。切割网格为具有嵌入边界的区域提供了一种便捷高效的网格生成手段，但由于其切割单元形状任意，可能难以使用。利用Chan的斜交混合SBP公式["斜对称熵稳定...", JSC, 2019]，我们提出了一种在切割网格上求解双曲守恒律的高阶精确、熵稳定格式。该公式要求在切割单元上使用正/非负权重的求积规则，我们通过显式参数化、子三角剖分和Caratheodory剪枝法来构造这些规则。我们使用浅水方程和可压缩欧拉方程数值验证了方法的精度和稳定性，并注意到将状态再分配技术与熵稳定方法结合使用所展现的积极前景。