Frequency optimized RBF-FD for wave equations

Abstract 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.

[1]  Bengt Fornberg,et al.  Stabilization of RBF-generated finite difference methods for convective PDEs , 2011, J. Comput. Phys..

[2]  Bengt Fornberg,et al.  A primer on radial basis functions with applications to the geosciences , 2015, CBMS-NSF regional conference series in applied mathematics.

[3]  C. Bogey,et al.  A family of low dispersive and low dissipative explicit schemes for flow and noise computations , 2004 .

[4]  Manuel Kindelan,et al.  Laurent series based RBF-FD method to avoid ill-conditioning , 2015 .

[5]  M. Moscoso,et al.  Optimized Finite Difference Formulas for Accurate High Frequency Components , 2016 .

[6]  Bengt Fornberg,et al.  Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids , 2005, Adv. Comput. Math..

[7]  Olav Holberg,et al.  COMPUTATIONAL ASPECTS OF THE CHOICE OF OPERATOR AND SAMPLING INTERVAL FOR NUMERICAL DIFFERENTIATION IN LARGE-SCALE SIMULATION OF WAVE PHENOMENA* , 1987 .

[8]  Manuel Kindelan,et al.  Optimal constant shape parameter for multiquadric based RBF-FD method , 2011, J. Comput. Phys..

[9]  Manuel Kindelan,et al.  Optimal variable shape parameter for multiquadric based RBF-FD method , 2012, J. Comput. Phys..

[10]  Manuel Kindelan,et al.  Laurent expansion of the inverse of perturbed, singular matrices , 2015, J. Comput. Phys..

[11]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[12]  Zhen-Xing Yao,et al.  Optimized explicit finite-difference schemes for spatial derivatives using maximum norm , 2013, J. Comput. Phys..

[13]  Yang Liu,et al.  Globally optimal finite-difference schemes based on least squares , 2013 .

[14]  Manuel Kindelan,et al.  RBF-FD formulas and convergence properties , 2010, J. Comput. Phys..