New RBF collocation schemes and their applications

Abstract

The purpose of this study is to apply some new RBF collocation schemes and recently-developed kernel RBFs to various types of partial differential equation systems. By analogy with the Fasshauer's Hermite interpolation, we recently developed the symmetric BKM and boundary particle methods (BPM), where the latter is based on the multiple reciprocity principle. The resulting interpolation matrix of them is always symmetric irrespective of boundary geometry and conditions. Furthermore, the proposed direct BKM and BPM apply the practical physical variables rather than expansion coefficients and become very competitive alternative to the boundary element method. On the other hand, by using the Green integral we derive a new domain-type symmetrical RBF scheme called as the modified Kansa method (MKM), which differs from the Fasshaure's scheme in that the MKM discretizes both governing equation and boundary conditions on the same boundary nodes. Therefore, the MKM significantly reduces calculation errors at nodes adjacent to boundary with explicit mathematical basis. Experimenting these novel RBF schemes with 2D and 3D Laplace, Helmholtz, and convection-diffusion problems will be subject of this study. In addition, the nonsingular high-order fundamental or general solution will be employed as the kernel RBFs in the BKM and MKM.