Solving an eigenvalue problem on a periodic domain using a radial basis function finite difference scheme

Local radial basis functions (RBFs) are becoming increasingly popular as an alternative to global RBFs, as the latter suffer from ill-conditioning. In this paper, a local meshless method based on RBFs in a finite-difference (FD) mode with better conditioned matrices has been developed for solving an eigenvalue problem with a periodic domain. Through numerical experiments, we examine the accuracy of the method as a result of variation in the number and layout of nodes in the domain and the effects of shape parameter, using various globally supported RBFs. The presented scheme has been validated on two different types of nodal arrangement, namely uniform and non-uniform node distributions. The results obtained from the method are found to be in good agreement with the benchmark analytical solutions. In addition, a higher-order RBF-FD scheme (which uses ideas from Hermite interpolation) is then proposed for solving the eigenvalue problem with a periodic domain. Tests show that both accuracy and convergence order can be improved dramatically by using higher-order RBF-FD formulae, which converge at a rate of O ( h 8.5 ) compared to the standard-order method which converges as O ( h 4.3 ) for uniformly distributed nodes with spacing h.