The Modified Localized Method of Approximated Particular Solutions for Linear and Nonlinear Convection-Diffusion-Reaction PDEs

Abstract

In this paper, a kernel based method, the modified localized method of approximated particular solutions (MLMAPS) [16, 23] is utilized to solve unsteady-state linear and nonlinear diffusion-reaction PDEs with or without convections. The time-space and spatial space are discretized by the higher-order Houbolt method with various time step sizes and the MLMAPS, respectively. The local truncation error associated with the time discretization is $\mathcal{O}(h^4)$, where $h$ is the largest time step size used. The spatial domain is then treated by a special kernel, the integrated polyharmonic splines kernels together with low-order polynomial basis. Typical computational algorithms require a trade off between accuracy and rate of convergency. However, the experimental analysis has shown high accuracy and fast convergence of the proposed method.