A spectral Galerkin method for nonlinear delay convection–diffusion–reaction equations

Abstract

This paper deals with the numerical approximation of a class of nonlinear delay convection–diffusion–reaction equations. In order to derive an efficient numerical scheme to solve the equations, we first convert the original equation into an equivalent reaction–diffusion problem with an exponential transformation. Then, we propose a fully discrete scheme by combining the Crank–Nicolson method and the Legendre spectral Galerkin method. The analytical and numerical stability criteria are obtained in L2-norm. It is proven under the suitable conditions that the method is convergent of second-order in time and of exponential order in space. Finally, several numerical experiments are given to illustrate the computational efficiency and the theoretical results.