Two-Dimensional RBF-ENO Method on Unstructured Grids
Two-Dimensional RBF-ENO Method on Unstructured Grids
复制标题
非结构化网格上的二维 RBF-ENO 方法
DOI：
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发表时间：
2020
影响因子：
2.5
通讯作者：
Fabian Mönkeberg
中科院分区：
文献类型：
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作者：
J. Hesthaven;Fabian Mönkeberg
等距笛卡尔网格上的本质非振荡 (ENO) 和加权 ENO (WENO) 方法广泛用于求解具有不连续解的偏微分方程。然而，非结构化网格上的稳定 ENO/WENO 方法研究较少。我们提出了一种基于径向基函数（RBF）的高阶 ENO 方法来求解一般二维网格上的双曲守恒定律。径向基函数重建提供了一种灵活的方法来处理病态细胞星座。我们引入了基于 RBF 的平滑度指标和适用于一般网格的模板选择算法。此外，我们开发了一种稳定的方法来评估有限体积设置中的 RBF 重建，该方法避免了误差停滞并保持重建的条件数有界。我们以几个具有挑战性的二维数值示例作为结论，以展示该方法的稳健性。
Essentially non-oscillatory (ENO) and weighted ENO (WENO) methods on equidistant Cartesian grids are widely used to solve partial differential equations with discontinuous solutions. However, stable ENO/WENO methods on unstructured grids are less well studied. We propose a high-order ENO method based on radial basis function (RBF) to solve hyperbolic conservation laws on general two-dimensional grids. The radial basis function reconstruction offers a flexible way to deal with ill-conditioned cell constellations. We introduce a smoothness indicator based on RBFs and a stencil selection algorithm suitable for general meshes. Furthermore, we develop a stable method to evaluate the RBF reconstruction in the finite volume setting which circumvents the stagnation of the error and keeps the condition number of the reconstruction bounded. We conclude with several challenging numerical examples in two dimensions to show the robustness of the method.