We derive stability estimates for three commonly used radial basis function (RBF) methods to solve hyperbolic time-dependent PDEs: the RBF generated finite difference (RBF-FD) method, the RBF partition of unity method (RBF-PUM) and Kansa's (global) RBF method. We give the estimates in the discrete -norm intrinsic to each of the three methods. The results show that Kansa's method and RBF-PUM can be -stable in time under a sufficiently large oversampling of the discretized system of equations. The RBF-FD method in addition requires stabilization of the spurious jump terms due to the discontinuous RBF-FD cardinal basis functions. Numerical experiments show an agreement with our theoretical observations.

中文翻译：

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应用于线性标量守恒定律的径向基函数方法的稳定性估计


我们推导了三种常用的径向基函数 (RBF) 方法的稳定性估计，以求解双曲时变偏微分方程：RBF 生成有限差分 (RBF-FD) 方法、RBF 统一划分方法 (RBF-PUM) 和 Kansa 的（全局）方法）径向基函数法。我们给出了三种方法中每种方法固有的离散范数的估计。结果表明，Kansa 方法和 RBF-PUM 在离散方程组足够大的过采样下能够保持时间稳定。由于 RBF-FD 基函数不连续，RBF-FD 方法还需要稳定寄生跳跃项。数值实验表明与我们的理论观察一致。