Numerical simulation of discretized second-order variable coefficient elliptic PDEs by a Classical Eight-step Model

Abstract

This article presents a new numerical model with nodes within the interval of eight-step for numerical simulation of a class of variable coefficient elliptic partial differential equations in the two-dimensional domain. The method is developed via the principle of collocation and interpolation techniques using Hermite polynomial as the basis function. The main classical model and its derivatives generated from the continuous function are then united together to form the required Classical Eight-step Model (CEM). The analysis of the CEM was investigated and shows that it satisfies the conditions for convergence with an algebraic order of nine. The CEM is applied to solve the semi-discretized elliptic partial differential equations with a variable coefficient arising from the discretization of one of their spatial variables. The performance of the CEM was established with six test problems. The approximate solution generated using CEM is compared with the analytical solution of the problems and other existing methods in the literature. Numerical illustrations demonstrate that the method is convergent and highly accurate. The advantages of CEM over other existing methods reveal the accuracy and efficiency of the CEM as depicted in curves.