A comparison of finite difference and finite volume methods for solving the space-fractional advection-dispersion equation with variable coefficients

Abstract

Transport processes within heterogeneous media may exhibit non-classical diffusion or dispersion which is not adequately described by the classical theory of Brownian motion and Fick's law. We consider a space-fractional advection-dispersion equation based on a fractional Fick's law. Zhang et al. [Water Resour. Res. 43:W05439, 2007] considered such an equation with variable coefficients, which they discretised using the finite difference method proposed by Meerschaert and Tadjeran [J. Comput. and Appl. Math. 172:65–77, 2004]. For this method, the presence of variable coefficients necessitates applying the product rule before discretising the Riemann–Liouville fractional derivatives using standard or shifted Grunwald formulas, depending on the fractional order. As an alternative, we propose using a finite volume method that deals directly with the equation in conservative form. Fractionally shifted Grunwald formulas are used to discretise the Riemann–Liouville fractional derivatives at control volume faces, eliminating the need for product rule expansions. We compare the two methods for several case studies, highlighting the convenience of the finite volume approach. References

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