A family of linearity-preserving schemes for anisotropic diffusion problems on arbitrary polyhedral grids. (English) Zbl 1286.76138
Summary: A family of cell-centered finite volume schemes are proposed for anisotropic diffusion problems on arbitrary polyhedral grids with planar facets. The derivation of the schemes is done under a general framework through a certain linearity-preserving approach. The key ingredient of our algorithm is to employ solely the so-called harmonic averaging points located at the cell interfaces to define the auxiliary unknowns, which not only makes the interpolation procedure for auxiliary unknowns simple and positivity-preserving, but also reduces the stencil of the schemes. The final schemes are cell-centered with a small stencil of 25-point on the structured hexahedral grids. Moreover, the schemes satisfy the local conservation condition, treat discontinuity exactly and allow for a simple stability analysis. A second-order accuracy in the \(L_2\) norm and a first-order accuracy in the \(H_1\) norm are observed numerically on general distorted meshes in case that the diffusion tensor is anisotropic and discontinuous.
MSC:
| 76R50 | Diffusion |
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 65N08 | Finite volume methods for boundary value problems involving PDEs |
Keywords:
anisotropic diffusion; cell-centered scheme; linearity-preserving criterion; harmonic averaging point; polyhedral gridsReferences:
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