The control-volume weighted flux scheme (CVWFS) for nonlocal diffusion and its relationship to fractional calculus

V. R. Voller, C. Paola, D. P. Zielinski

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In diffusion transport, the flux at a point is typically modeled in terms of the local gradient of a potential. When heterogeneities are present, this local model can break down and it may be more appropriate to model the diffusion flux as a weighted sum of gradients present throughout the domain. Here a discrete nonlocal flux modelconsistent with control-volume implementations-is developed. This scheme is referred to as the control-volume weighted flux scheme (CVWFS). The key component is the modeling of the diffusion flux at a given control-volume face in terms of a weighted sum of gradients at that face and at faces up- and downstream. Criteria for choosing the weights are proposed. This results in numerical solution schemes in which the coefficient matrix is diagonally dominant, has positive off-diagonal elements, and zero row sums. For a particular power-law weighting scheme it is shown how the CVWFS is related to the definition of the Caputo fractional derivative and the one-shift Grunwald approximation of the Riemann-Liouville fractional derivative. On developing transients and boundary condition treatments, the accuracy and suitability of the CVWFS scheme is demonstrated by solving a number of problems governed by Caputo fractional diffusion equations.

Original languageEnglish (US)
Pages (from-to)421-441
Number of pages21
JournalNumerical Heat Transfer, Part B: Fundamentals
Issue number6
StatePublished - Jun 2011

Bibliographical note

Funding Information:
Received 23 August 2010; accepted 11 March 2011. This work was supported by the STC program of the National Science Foundation via the National Center for Earth-surface Dynamics under agreement EAR-0120914. The authors are grateful to Gary Bohannan for reading manuscript drafts. Address correspondence to Vaughan R. Voller, Department of Civil Engineering, University of Minnesota, 122 Civil Engineering Building, 500 Pillsbury Drive SE, Minneapolis, MN 55455-0116, USA. E-mail:


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