The Space–Time Kernel-Based Numerical Method for Burgers’ Equations

Abstract

It is well known that major error occur in the time integration instead of the spatial approximation. In this work, anisotropic kernels are used for temporal as well as spatial approximation to construct a numerical scheme for solving nonlinear Burgers’ equations. The time-dependent PDEs are collocated in both space and time first, contrary to spatial discretization, and time stepping procedures for time integration are then applied. Physically one cannot in general expect that the spatial and temporal features of the solution behaves on the same order. Hence, one should have to incorporate anisotropic kernels. The nonlinear Burgers’ equations are converted by nonlinear transformation to linear equations. The spatial discretizations are carried out to construct differentiation matrices. Comparisons with most available numerical methods are made to solve the Burgers’ equations.