在复杂几何的点云上构造稳定的高阶有限差分算子
Constructing stable, high-order finite-difference operators on point clouds over complex geometries
AI总结：
提出一种在水平集几何点云上构建高阶对角范数一阶导数SBP算子的算法，利用临时笛卡尔切割单元网格定义稀疏模式与求积规则，通过内点法计算对角范数，并用线性平流方程验证了精度与稳定性。
摘要：
具有求和部分（SBP）性质的高阶差分算子可用于构建双曲守恒律的稳定离散化；然而，大多数高阶SBP算子需要针对感兴趣区域使用一致的高阶网格。为规避这一要求，我们提出了一种算法，用于在水平集几何的点云上构建高阶、对角范数、一阶导数SBP算子。该算法并非无网格方法，因为它使用笛卡尔切割单元网格来定义算子的稀疏模式并提供中间求积规则；然而，该网格是自动生成的，并且一旦SBP算子构建完成即可丢弃。利用这一临时网格，我们构造局部的、基于单元的SBP差分算子，并将其组装成全局SBP算子。我们确定了正定对角质量矩阵存在的条件，并通过使用内点算法求解线性不等式的稀疏系统来计算对角范数。我们还描述了一种人工耗散算子，在求解双曲问题时作为一阶导数算子的补充，尽管该耗散对于稳定性并非必需。数值结果证实了对角范数存在的条件，并研究了范数条目的分布。此外，结果使用线性平流方程验证了点云SBP算子的精度和稳定性。
英文摘要：
High-order difference operators with the summation-by-parts (SBP) property can be used to build stable discretizations of hyperbolic conservation laws; however, most high-order SBP operators require a conforming, high-order mesh for the domain of interest. To circumvent this requirement, we present an algorithm for building high-order, diagonal-norm, first-derivative SBP operators on point clouds over level-set geometries. The algorithm is not mesh-free, since it uses a Cartesian cut-cell mesh to define the sparsity pattern of the operators and to provide intermediate quadrature rules; however, the mesh is generated automatically and can be discarded once the SBP operators have been constructed. Using this temporary mesh, we construct local, cell-based SBP difference operators that are assembled into global SBP operators. We identify conditions for the existence of a positive-definite diagonal mass matrix, and we compute the diagonal norm by solving a sparse system of linear inequalities using an interior-point algorithm. We also describe an artificial dissipation operator that complements the first-derivative operators when solving hyperbolic problems, although the dissipation is not required for stability. The numerical results confirm the conditions under which a diagonal norm exists and study the distribution of the norm's entries. In addition, the results verify the accuracy and stability of the point-cloud SBP operators using the linear advection equation.
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