A space-time pseudospectral method for solving multi-dimensional quasi-linear parabolic partial differential (Burgers') equations

Abstract

The viscous Burgers' equation is a kind of quasi-linear important partial differential equation appeared in many fields. The nonlinear term contained in this equation results in difficulty in finding its high-precision numerical solution. In this paper, a space-time pseudospectral method for solving multi-dimensional Burgers' equations is proposed based on the Lagrange polynomials. The method is employed in time and space both at Chebyshev- Gauss- Lobbato(CGL) points. The spectral coefficients are found in such a way that the residual becomes minimum. The given problem is reduced to a system of nonlinear algebraic equations, which is solved by Newton-Raphson's method. Error bounds on discrete L2-norm and Sobolev norm (Hp) are presented. The computational experiments are carried out to corroborate the theoretical results and to compare the present method with existing methods in the literature. Moreover, some model examples of multi-dimensional Burgers' equations are tested and a detailed comparison of the proposed method with various other methods is also given. When using the proposed method, some interesting phenomenon appears with the coefficient of kinematic viscous.