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 $\ell_2$-norm intrinsic to each of the three methods.The results show that Kansa's method and RBF-PUM can be $\ell_2$-stable in time under a sufficiently large oversampling of the discretized system of equations. On the other hand, the RBF-FD method is not $\ell_2$-stable by construction, no matter how large the oversampling is. We show that this is due to the jumps (discontinuities) in the RBF-FD cardinal basis functions. We also provide a stabilization of the RBF-FD method that penalizes the spurious jumps. Numerical experiments show an agreement with our theoretical observations.

翻译：我们得出三种常用的双曲基函数(RBF)的稳定性估计值,以解决双曲时间依赖的PDE:RBF产生的有限差异(RBF-FD)方法、RBF的统一方法(RBF-PUM)和Kansa的(Global)RBF方法(RBF)的稳定性估计值。我们给出了三种方法中每种方法所固有的离散 $@ ell_ 2$-norm的估计数。结果显示,Kansa 的方法和RBF-PUM在足够大地过度地取样离散式方程式系统的情况下,在时间上可以达到$\ ell_ 2$- sable。另一方面,RBFF-FD方法不是按构造来测得的 $\ ell_ 2$- sable- sable, 不论过度采样大小。我们表明,这是由于RBFFFFF-FFM基础功能中的跳跃(不连续) 造成的。我们还提供了一种稳定RBFFFFFD-FD方法,以惩罚刺激性跳跃式跳。Numerical 实验表明我们同意。