A reduced order finite difference method for solving space-fractional reaction-diffusion systems: The Gray-Scott model

In this paper, we want to present a fast, efficient, and robust numerical procedure for solving a system of PDEs with regard to the fractional Laplacian equation. In the developed approximate scheme, the spatial direction is discretized by a second-order finite difference formula and the temporal direction is discretized by a first-order finite difference approximation. In order to improve the accuracy and to decrease the used CPU time an alternative direction implicit (ADI) method is employed. Furthermore, we use a reduced order model (ROM) based on the proper orthogonal decomposition (POD) technique to reduce the elapsed computational time. The developed numerical scheme is well known as the reduced order finite difference scheme. To emphasize the fast and efficiency of the proposed algorithm, we apply it for the two-dimensional case.