Keywords: Caputo derivative, fractional diffusion, two-dimensions.

In some diffusion processes, as a result of the presence of heterogeneities, the time exponent for the spreading length scale of an initial pulse can differ from the classical one-half; such a process is referred to as anomalous diffusion. A standard model for this non-local diffusive transport, valid when the length scales of the heterogeneities are power-law distributed, is to represent the flux in terms of a fractional derivative. Recently Voller et al. [1] obtained a solution to the one-dimensional fractional diffusion equation using a discrete control volume approach called the control volume weighted flux scheme (CVWFS). An approach that can be viewed as an alternative to methods based on the L1/L2 [2,3] or one-shift Grünwald [4,5] approximations of the fractional diffusion flux. Here, the CVWFS is expanded and numerical solutions developed for two-dimensional transient Caputo fractional diffusion equations. Where possible this approach was verified using available analytical solutions.

The main contributions of this paper are:

1. An explicit test of the relative accuracy of the CVWFS using the originally proposed weights in Voller et al. [1].
2. A demonstration that the CVWFS can also operate with weights derived from previous literature schemes [2,3,4,5].
3. An extension, with verification, of the CVWFS to the solution of two-dimensional transient and steady-state fractional diffusion equations.

This paper clearly demonstrates that the accuracy of the CVWFS using the originally proposed weights is of the same order as the scheme operating with the Grünwald weights. Further, for a full range of fractional orders, including different values in the coordinate directions, solutions of the extended two-dimensional CVWFS are in good agreement with available analytical solutions.

1

V.R. Voller, C. Paola, D.P. Zielinski, "The Control Volume Weighted Flux Scheme (CVWFS) for Non-Local Diffusion and Its Relationship to Fractional Calculus", Numerical Heat Transfer B, 59, 421-441, 2011.

2

K.B. Oldham, J. Spanier, "The fractional calculus", Academic Press, New York, USA, 1974.

3

V.E. Lynch, B.A. Carreras, D. del-Castillo-Negrete, K.M. Ferreira-Mejias, H.R. Hicks, "Numerical methods for the solution of partial differential equations of fractional order", Journal of Computational Physics, 192, 406-421, 2003. doi:10.1016/j.jcp.2003.07.008

4

M.M. Meerschaert, H. Scheffler, C. Tadjeran, "Finite difference methods for two-dimensional fractional dispersion equation", Journal of Computational Physics, 211, 249-261, 2006. doi:10.1016/j.jcp.2005.05.017

5

Y. Zhang, D. Benson, M.M. Meerschaert, E.M. LaBolle, "Space-fractional advection-dispersion equations with variable parameters: Diverse formulas, numerical solutions, and application to the Macrodispersion Experiment site data", Water Resources Research, 43, W05439, 2007. doi:10.1029/2006WR004912

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