Interpolation-based second-order monotone finite volume schemes for anisotropic diffusion equations on general grids. (English) Zbl 1349.65586
Summary: 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.
MSC:
| 65N08 | Finite volume methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
Keywords:
diffusion equation; monotone finite volume scheme; linearity-preserving criterion; nonlinear two-point flux approximationSoftware:
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