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

Hejazi, Hala, Moroney, Tim, & Liu, Fawang (2013) A comparison of finite difference and finite volume methods for solving the space-fractional advection-dispersion equation with variable coefficients. The ANZIAM Journal, 54, C557-C573.

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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 Resources Research, 43(5)(2007)] considered such an equation with variable coefficients, which they dis- cretised using the finite difference method proposed by Meerschaert and Tadjeran [Journal of Computational and Applied Mathematics, 172(1):65-77 (2004)]. For this method the presence of variable coef- ficients necessitates applying the product rule before discretising the Riemann–Liouville fractional derivatives using standard and shifted Gru ̈nwald 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 Gru ̈nwald 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.

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ID Code: 69063
Item Type: Journal Article
Refereed: Yes
Additional URLs:
Keywords: finite volume, finite difference, space-fractional advection-dispersion equation, Fractionally-shifted Grunwald formulas
ISSN: 1446-8735
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > NUMERICAL AND COMPUTATIONAL MATHEMATICS (010300) > Numerical Solution of Differential and Integral Equations (010302)
Divisions: Current > Schools > School of Mathematical Sciences
Current > QUT Faculties and Divisions > Science & Engineering Faculty
Copyright Owner: Copyright 2013 Australian Mathematical Society
Deposited On: 23 Mar 2014 22:36
Last Modified: 30 Mar 2016 09:15

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