Differential Quadrature Method for Two-Dimensional Burgers' Equations

Abstract

In this paper we propose a rapid convergent differential quadrature method (DQM) for calculating the numerical solutions of nonlinear two-dimensional Burgers' equations with appropriate initial and boundary conditions. The two dimensional Burgers' equations arise in various kinds of phenomena such as a mathematical model of turbulence and the approximate theory of flow through a shock wave traveling in a viscous fluid. To the best of the authors' knowledge, this is the first attempt that the system has been solved up to big time level t = 15 and big Reynolds number R = 1200. We also found that Chebyshev-Gauss-Lobatto grid points give excellent results in comparison to other grid points such as uniform grid points. Two test problems considered by different researchers have been studied to demonstrate the accuracy and utility of the present method. Results obtained by the method have been compared with the exact solutions and with those given by researchers for each problem. Solutions obtained by DQM are found to be very good and useful. Convergence and stability of the method is also examined.