Abstract

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.

中文翻译：

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非结构化网格上的二维RBF-ENO方法

摘要

等距笛卡尔网格上的本质上非振荡（ENO）和加权ENO（WENO）方法被广泛用于求解具有不连续解的偏微分方程。但是，对非结构化网格上稳定的ENO / WENO方法的研究较少。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律。径向基函数重建提供了一种处理病态细胞星座的灵活方法。我们介绍了一种基于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，从而避免了错误的停滞并保持重建的条件数有界。