New RBF collocation schemes and their applications

TL;DR

This paper introduces new RBF collocation schemes, including symmetric BKM, BPM, and MKM, for solving PDEs, demonstrating their advantages in symmetry, boundary accuracy, and application to various 2D and 3D problems.

Contribution

The study develops novel symmetric RBF schemes and boundary methods that improve accuracy and symmetry, offering practical advantages over existing boundary element methods.

Findings

Symmetric BKM and BPM produce symmetric matrices regardless of boundary shape.

MKM reduces boundary node errors by discretizing boundary conditions directly.

High-order fundamental solutions enhance the accuracy of RBF kernels.

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.