Predictor-Corrector Scheme for Multi-Dimensional Time-Space Fractional Reaction-Diffusion Models

A computationally efficient and highly accurate predictor-corrector numerical scheme is developed to solve multi-dimensional time-space fractional reaction diffusion models. The fractional Laplacian is discretized using the matrix transfer technique based on the fourth-order compact finite difference method. A predictor-corrector time-stepping scheme is developed using the fractional Euler method as a predictor and the fractional Adams method as a corrector. The accuracy of the scheme is further enhanced by applying the Richardson extrapolation. Utilizing the discrete Fourier transform, a computationally efficient algorithm is developed for the straightforward implementation of the scheme to one- and multi-dimensional problems. The dynamics of the solution due to the memory effect and/or anomalous diffusion are demonstrated through numerical experiments. Graphs of the solution profiles for two- and three-dimensional problems are presented. The computational accuracy and efficiency of the presented scheme are verified by computing the convergence rates and the central processing unit time.