Optimization of finite difference method with multiwavelet bases

Abstract

Multiwavelet based methods are among the latest techniques to solve partial differential equations (PDEs) numerically. Finite Difference Method (FDM) - powered by its simplicity - is one of the most widely used techniques to solve PDEs numerically. But it fails to produce better result in problems where the solution is having both sharp and smooth variations at different regions of interest. In such cases, to achieve a given accuracy an adaptive scheme for proper grid placement is needed. In this paper we propose a method, 'Multiwavelet Optimized Finite Difference Method' (MWOFD) to overcome the drawback of FDM. In the proposed method, multiwavelet coefficients are used to place non-uniform grids adaptively. This method is highly converging and requires only less number of grids to achieve a given accuracy when contrasted with FDM. The method is demonstrated for nonlinear Schrödinger equation and Burgers' equation. © 2009 Springer Berlin Heidelberg.