Modified Finite Integration Method Using Chebyshev Polynomial Expansion for Solving One-Dimensional Nonlinear Burgers' Equations with Shock Wave

Ampol Duangpan, Ratinan Boonklurb

Abstract

Based on the recently modified finite integration method (FIM) for solving linear differential equations by using the Chebyshev polynomial expansion, in this paper, we improve the modified FIM to be able to handle nonlinear Burgers' equations with shock waves in one dimension. The main idea is to approximate the nonlinear term of the Burgers' equation and apply the modified FIM to construct the finite integration matrices on each computational grid points which are generated by the zeros of the Chebyshev polynomial of a certain degree. In addition, the term involving partial derivative with respect to time is approximated by the forward difference quotient. Illustrative numerical solutions obtained by the proposed modified FIM algorithm are compared with the traditional FIM, finite difference method (FDM), finite element method (FEM), other methods and their analytical solution from several examples. Evidently, the proposed modified FIM algorithm has made a significant improvement in terms of accuracy and computational time for small values of the viscosity.