Uniform convergence of compact and BDF methods for the space fractional semilinear delay reaction–diffusion equations

Abstract

In this article, two classes of finite difference methods are constructed to solve the space fractional semilinear delay reaction–diffusion equations. Firstly, fractional centered finite difference method in space and backward differential formula (BDF2) in time are employed to discrete the original equations. Secondly, the existence, uniqueness and convergence of the numerical method are analyzed at length by G-norm technique, and the convergence order is O(τ2+h2). Then the compact technique of finite difference method is used to further increase the accuracy in spatial direction, and a method with fourth accuracy in spatial dimension is obtained. The convergence orders are proved to be O(τ2+h4) in the sense of three classes of norms by GA-norm technique. Especially, we obtain the uniform convergence for the error estimation, which demonstrates competitive performance compared with the preceding work in the related references. Finally, two numerical examples are provided to verify our theoretical results and demonstrate the effectiveness of both methods when applied to simulate space fractional delay diffusive Nicholson’s blowflies equation.