Efficient Chebyshev Pseudospectral Methods for Viscous Burgers’ Equations in One and Two Space Dimensions

Abstract

Spectral methods are powerful numerical methods used for the solution of ordinary and partial differential equations. In this paper, we propose an efficient and accurate numerical method for the one and two dimensional nonlinear viscous Burgers’ equations and coupled viscous Burgers’ equations with various values of viscosity subject to suitable initial and boundary conditions. The method is based on the Chebyshev collocation technique in space and the fourth-order Runge–Kutta method in time. This proposed scheme is robust, fast, flexible, and easy to implement using modern mathematical software such as Matlab. Furthermore, it can be easily modified to handle other problems. Extensive numerical simulations are presented to demonstrate the accuracy of the proposed scheme. The numerical solutions for different values of the Reynolds number are compared with analytical solutions as well as other numerical methods available in the literature. Compared to other numerical methods, the proposed method is shown to have higher accuracy with fewer nodes. Our numerical experiments show that the Chebyshev collocation method is an efficient and reliable scheme for solving Burgers’ equations with reasonably high Reynolds number.