A hybrid DQ/LMQRBF-DQ approach for numerical solution of Poisson-type and Burger’s equations in irregular domain

Abstract

Differential quadrature (DQ) is an efficient and accurate numerical method for solving partial differential equations (PDEs). However, it can only be used in regular domains in its conventional form. Local multiquadric radial basis function-based differential quadrature (LMQRBF-DQ) is a mesh free method being applicable to irregular geometry and allowing simple imposition of any complex boundary condition. Implementation of the latter numerical scheme imposes high computational cost due to the necessity of numerous matrix inversions. It also suffers from sensitivity to shape parameter(s). This paper presents a new method through coupling the conventional DQ and LMQRBF-DQ to solve PDEs. For this purpose, the computational domain is divided into a few rectangular shapes and some irregular shapes. In such a domain decomposition process, a high percentage of the computational domain will be covered by regular shapes thus taking advantage of conventional DQM eliminating the need to implement Local RBF-DQ over the entire domain but only on a portion of it. By this method, we have the advantages of DQ like simplicity, high accuracy, and low computational cost and the advantages of LMQRBF-DQ like mesh free and Dirac’s delta function properties. We demonstrate the effectiveness of the proposed methodology using Poisson and Burgers’ equations.