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ABSTRACT
The focus of this paper is the numerical solution of space-fractional heat conduction equations. Typical numerical treatments for fractional diffusion equations (FDEs) are constructed to function on structured grids and cannot be readily applied to problems in arbitrarily shaped domains. Presented is an unstructured control volume finite element method for the solution of FDEs in non-Cartesian domains. This method approximates the fractional derivative flux at given integration points with a weighted sum of gradients along orthogonal spines. Accuracy and utility of the approach are demonstrated by solution of one- and two-sided FDEs in rectangular and cylindrical domains.
Nomenclature
| α | = | fractional order or locality |
| β | = | bias weighting |
| Γ | = | gamma function |
| δ | = | maximum side length of control volume |
| Δt | = | time step |
| Δx, Δy | = | space step |
| ν | = | diffusivity |
| g | = | Grünwald weight |
| G | = | Gaussian weight |
| L | = | length of spine |
| NS | = | number of steps along spine |
| q | = | flux |
| qα | = | fractional flux |
| Q | = | flow across control volume face |
| S | = | shape functions |
| T | = | local temperature |
| V | = | control volume area |
| W | = | flux weight |
| x, y | = | space coordinate |
| = | fractional derivative from left and right | |
| Subscript | = | |
| x, y | = | coordinate axis |
| Superscript | = | |
| RL | = | Riemann–Liouville derivative |
Nomenclature
| α | = | fractional order or locality |
| β | = | bias weighting |
| Γ | = | gamma function |
| δ | = | maximum side length of control volume |
| Δt | = | time step |
| Δx, Δy | = | space step |
| ν | = | diffusivity |
| g | = | Grünwald weight |
| G | = | Gaussian weight |
| L | = | length of spine |
| NS | = | number of steps along spine |
| q | = | flux |
| qα | = | fractional flux |
| Q | = | flow across control volume face |
| S | = | shape functions |
| T | = | local temperature |
| V | = | control volume area |
| W | = | flux weight |
| x, y | = | space coordinate |
| = | fractional derivative from left and right | |
| Subscript | = | |
| x, y | = | coordinate axis |
| Superscript | = | |
| RL | = | Riemann–Liouville derivative |