Cartesian grid methods using radial basis functions for solving Poisson, Helmholtz, and diffusion–convection equations

Abstract

Four methods that solve the Poisson, Helmholtz, and diffusion–convection problems on Cartesian grid by collocation with radial basis functions are presented. Each problem is split into a problem with an inhomogeneous equation and homogeneous boundary conditions, and a problem with a homogeneous equation and inhomogeneous boundary conditions. The former problem is solved by collocation with multiquadrics, whereas the latter problem is solved by collocation with either multiquadrics or fundamental solutions. It is found that methods that make use of fundamental solutions for collocation yield more accurate solutions that are less sensitive to the shape parameter of multiquadrics and node arrangement. Additional collocation appears to improve the quality of solutions.