Fast hybrid explicit group methods for solving 2D fractional advection-diffusion equation

In recent years, fractional partial differential equations (FPDEs) have been viewed as powerful mathematical tools for describing ample phenomena in various scientific disciplines and have been extensively researched. In this article, the hybrid explicit group (HEG) method and the modified hybrid explicit group (MHEG) method are proposed to solve the 2D advection-diffusion problem involving fractional-order derivative of Caputo-type in the temporal direction. The considered problem models transport processes occurring in real-world complex systems. The hybrid grouping methods are developed based upon a Laplace transformation technique with a pair of explicit group finite difference approximations constructed on different grid spacings. The proposed methods are beneficial in reducing the computational burden resulting from the nonlocality of fractional-order differential operator. The theoretical investigation of stability and convergence properties is conducted by utilizing the matrix norm analysis. The improved performance of the proposed methods against a recent competitive method in terms of central processing unit (CPU) time, iterations number and computational cost is illustrated by several numerical experiments.