Using a shifted Grünwald‐Letnikov scheme for the Caputo derivative to study anomalous solute transport in porous medium

Abstract

SummaryWe propose an extension of the shifted Grünwald‐Letnikov method to solve fractional partial differential equations in the Caputo sense with arbitrary fractional order derivative α and with an advective term. The method uses the relation between Caputo and Riemann‐Liouville definitions, the shifted Grünwald‐Letnikov, and the traditional backward and forward finite difference method. The stability of the method is investigated for the implicit and explicit scheme with homogeneous boundary conditions, and a stability criterion is found for the advective‐dispersive equation. An application of the method is used to solve contaminant diffusion and advective‐dispersive problems. The numerical solution for the fractional diffusion and fractional advection‐dispersion is compared with their respective analytical solutions for different time and space grid refinements. The diffusion simulation exhibited a good fit between the analytical and numerical solutions, with the explicit scheme going from stable to unstable as the time and space refinement changes. The fractional advection‐dispersion application produced small deviations from the analytical solution. These deviations, however, are analogous to the numerical dispersions encountered in conventional finite difference solutions of the advection‐dispersion equation. The new method is also compared with the traditional L2 method. Notably, an example that involves asymmetrical fractional conditions, a fractional diffusivity that depends on time, and a source term show how the methods compare. Overall, this study assesses the quality and easiness of use of the numerical method.