A computational approach for the space-time fractional advection–diffusion equation arising in contaminant transport through porous media

Abstract

The fractional advection–diffusion equation, known as non-local diffusion, is a relationship utilized in groundwater hydrology as a reliable means of modeling the transport of passive tracers in porous media by fluid flow. The main target of this paper is to develop a numerical method for solving the space-time fractional advection–diffusion equation (STFADE) defined by Caputo sense. In this way, a finite difference formula with first-order accuracy is used to discretize the problem in the temporal direction, and also the Chebyshev collocation method of the third kind is applied to approximate the space variable. The stability and convergence of the fully discrete scheme are rigorously established in $$L^{2}$$ norm. The numerical results are presented and are compared with other methods to show the capability of the numerical scheme proposed here.