Frequency optimized RBF-FD for wave equations

We present a method to obtain optimal RBF-FD formulas which maximize their frequency range of validity. The optimization is based on the idea of keeping an error of interest (dispersion, phase or group velocity errors) below a given threshold for a wavenumber interval as large as possible. To find the weights of these optimal finite difference formulas we solve an optimization problem. In a previous work we developed a method to optimize the frequency range of validity for finite difference weights. That method required to solve a system of nonlinear equations with as many unknowns as half of the number of weights, which is a very hard task when the number of nodes gets large. The current method requires solving an optimization problem with only one parameter, which makes finding a global minimum easier, and thus can be used for bigger stencils. We also study which of the standard RBF are more appropriate for this problem and introduce a new RBF that depends on two parameters. This new RBF improves the resulting frequency response of the RBF-FD methods while keeping the cost of the optimization problem low.