A new space–time localized meshless method based on coupling radial and polynomial basis functions for solving singularly perturbed nonlinear Burgers’ equation

Abstract

In this paper, the singularly perturbed nonlinear Burgers’ problem (SPBP) with small kinematic viscosity $0<ϵ\ll 1$ is solved using a new Space–Time Localized collocation method based on coupling Polynomial and Radial Basis Functions (STLPRBF). To our best knowledge, it is the first time that the solution of SPBP is accurately approximated using the space–time meshless method without applying any adaptive refinement technique. The method is based on solving the problem without distinguishing between space and time variables, which eliminates the need for time discretization schemes. To address the inherent non-linearity of the problem, the method employs an iterative algorithm based on quasilinearization technique. The efficiency and accuracy of the proposed method are demonstrated by solving different examples of one- and two-dimensional SPBP with very small $ϵ$ up to $10^{-10}$. Additionally, the numerical convergence of the method with respect to $ϵ$ and also to the number of collocation points has been investigated. The comparison of the STLPRBF results with other published ones is presented.