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Interpolation-based second-order monotone finite volume schemes for anisotropic diffusion equations on general grids

Abstract : We propose two interpolation-based monotone schemes for the anisotropic diffusion problems on unstructured polygonal meshes through the linearity-preserving approach. The new schemes are characterized by their nonlinear two-point flux approximation, which is different from the existing ones and has no constraint on the associated interpolation algorithm for auxiliary unknowns. Thanks to the new nonlinear two-point flux formulation, it is no longer required that the interpolation algorithm should be a positivity-preserving one. The first scheme employs vertex unknowns as the auxiliary ones, and a second-order but not positivity-preserving interpolation algorithm is utilized. The second scheme uses the so-called harmonic averaging points located on cell edges to define the auxiliary unknowns, and a second-order positivity-preserving interpolation method is employed. Both schemes have nearly the same convergence rates as compared with their related second-order linear schemes. Numerical results demonstrate that {the new} schemes are monotone, and have the second-order accuracy for the solution and first-order for its gradient on severely distorted meshes.
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https://hal.inria.fr/hal-00907299
Contributor : Zhiming Gao <>
Submitted on : Monday, June 9, 2014 - 4:27:35 AM
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Jiming Wu, Zhiming Gao. Interpolation-based second-order monotone finite volume schemes for anisotropic diffusion equations on general grids. 2013. ⟨hal-00907299v2⟩

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