An effective operator splitting method based on spectral deferred correction for the fractional Gray–Scott model

Abstract

This paper presents a method by combining the semi-implicit spectral deferred correction (SDC) method with the operator splitting scheme to simulate the fractional Gray-Scott (GS) model. We start with the second-order operator splitting scheme, which is to split the original problem into linear and nonlinear parts. The linear subproblem is numerically solved using the Fourier spectral method, which is based on the exact solution and thus has no stability restriction on the time-step size. The nonlinear subproblem is solved via the Crank-Nicolson formula and Rubin-Graves linearization technique, which can be solved effectively. The stability and convergence of this method are analyzed in L2-norm. Moreover, the scheme also takes advantage of the semi-implicit SDC method to improve the temporal accuracy. Numerical results are given to illustrate that the proposed method is a practical, accurate and efficient simulation tool for solving fractional GS problems.