A linear scheme satisfying a maximum principle for anisotropic diffusion operators on distorted grids. (Un schéma linéaire vérifiant le principe du maximum pour des opérateurs de diffusion très anisotropes sur des maillages déformés.) (French. Abridged English version) Zbl 1158.65079
Summary: We introduce a generalized finite-difference method for anisotropic diffusion operators on distorted grids. We calculate the second-order derivatives in space using a Taylor expansion. The resulting global matrix associated to the scheme is an M-matrix. Thanks to a certain assumption on the grid properties, we show the convergence of the scheme. We show the robustness of the method in comparison with analytical solutions and results obtained by other numerical schemes.
MSC:
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
Keywords:
numerical examples; finite volume method; finite-difference method; anisotropic diffusion operators; distorted grids; M-matrix; convergenceReferences:
| [1] | Bernard-Michel, G.; Le Potier, C.; Beccantini, A.; Gounand, S.; Chraibi, M., The Andra Couplex1 test: comparisons between finite element, mixed hybrid finite element and finite volume element discretizations: simulation of transport around a nuclear waste disposal site, Comput. Geosci., 8, 2, 187-201 (2004) · Zbl 1056.86001 |
| [2] | Burman, E.; Ern, A., Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes, C. R. Acad. Sci. Paris, Ser. I, 338, 8, 641-646 (2004) · Zbl 1049.65128 |
| [3] | F. Dabbène, Mixed hybrid finite elements for transport of polluants by undergrounds water, in: Proceeding of the 10th International Conference on Finite Elements in Fluids, Tucson, USA, 1998; F. Dabbène, Mixed hybrid finite elements for transport of polluants by undergrounds water, in: Proceeding of the 10th International Conference on Finite Elements in Fluids, Tucson, USA, 1998 |
| [4] | Dantzig, G. B., Maximization of a linear function of variables subject to linear inequalities, (Koopmans, T. C., Activity Analysis of Production and Allocation (1951), Wiley: Wiley New York), 339-347, (Chapter 21) · Zbl 0045.09802 |
| [5] | Eymard, R.; Gallouët, T.; Herbin, R., Finite volume method, (Ciarlet, P. G.; Lions, J. L., Handbook of Numerical Analysis, vol. VII (2000)) · Zbl 0843.65068 |
| [6] | I. Faille, Modélisation bidimensonnelle de la génèse et de la migration des hydrocarbures dans un bassin sédimentaire, Thèse de l’université Joseph Fourier-Grenoble 1, 1992; I. Faille, Modélisation bidimensonnelle de la génèse et de la migration des hydrocarbures dans un bassin sédimentaire, Thèse de l’université Joseph Fourier-Grenoble 1, 1992 |
| [7] | Gavete, L.; Gavete, M. L.; Benito, J. J., Improvements of generalized finite difference method and comparison with other meshless method, Appl. Math. Modelling, 27, 10, 831-847 (2003) · Zbl 1046.65085 |
| [8] | A. Genty, C. Le Potier, Avoiding negative concentrations for transport calculations of radionuclides in high-level radioactive waste disposal system, in: IAHR, International Groundwater Symposium, Istanbul, 2008, pp. 286-292; A. Genty, C. Le Potier, Avoiding negative concentrations for transport calculations of radionuclides in high-level radioactive waste disposal system, in: IAHR, International Groundwater Symposium, Istanbul, 2008, pp. 286-292 |
| [9] | Herbin, R.; Hubert, F., Benchmark on discretization schemes for anisotropic diffusion problems on general grids (2008), in: 5th International Symposium on Finite Volumes for Complex Applications, June 08-13 |
| [10] | Le Potier, C., Finite volume scheme for highly anisotropic diffusion operators on unstructured meshes, C. R. Acad. Sci. Paris, Ser. I, 340, 921-926 (2005) · Zbl 1076.76049 |
| [11] | Le Potier, C., Finite volume monotone scheme for highly anisotropic diffusion operators on unstructured triangular meshes, C. R. Acad. Sci. Paris, Ser. I, 341, 787-792 (2005) · Zbl 1081.65086 |
| [12] | C. Le Potier, Finite volume scheme satisfying maximum and minimum principles for anisotropic diffusion operators, in: Finite Volumes for Complex Applications V, Aussois, 2008, pp. 103-118; C. Le Potier, Finite volume scheme satisfying maximum and minimum principles for anisotropic diffusion operators, in: Finite Volumes for Complex Applications V, Aussois, 2008, pp. 103-118 · Zbl 1374.76155 |
| [13] | Xu, J.; Zikatanov, L., A monotone finite element scheme for convection-diffusion equations, Math. Comp., 66, 228, 1429-1446 (1999) · Zbl 0931.65111 |
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