A numerically robust and stable time–space pseudospectral approach for multidimensional generalized Burgers–Fisher equation

Abstract

We study the pseudospectral discretization of a nonlinear multidimensional generalized Burgers–Fisher equation. The main objective of this study is to establish that the implementation of the pseudospectral method in time holds equal importance to its application in space when addressing nonlinearity. Typically, the literature recommends using a large number of grid points in the temporal direction to ensure satisfactory outcomes. However, the fact that we obtained excellent accuracy with a relatively small number of temporal grid points is a notable finding in this study. The Chebyshev–Gauss–Lobatto (CGL) points serve as the foundation for the recommended method. By applying the proposed method to the problem, a system of algebraic equations is obtained, which is subsequently solved using the Newton–Raphson technique. We have conducted stability analysis and error estimation for the proposed method. Different researchers’ considerations on test problems have been explored to illustrate the robustness and practicality of the approach presented. The results obtained using the proposed method demonstrate a high level of accuracy, surpassing the existing solutions by a significant margin.