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Quasi-positive families of continuous Darcy-flux finite volume schemes on structured and unstructured grids.(English)Zbl 1402.76078

Summary: New families of flux-continuous control-volume distributed finite volume schemes are presented for the general full-tensor pressure equation arising in porous media and formulated for structured and unstructured grids. These schemes offer the practical advantage of being flux-continuous while only depending on one degree of freedom per control-volume, unlike rival approximations such as the Mixed Finite Element method. \(M\)-matrix bounds are presented, quasi QM-matrices are defined and an optimal quadrilateral scheme is identified. Anisotropy favoring triangulation is also shown to yield an optimal scheme. The new schemes prove to be relatively robust for the cases tested, including strongly anisotropic full tensor fields. Strong oscillations encountered with the earlier formulations, are removed or minimized.

MSC:

76M12 Finite volume methods applied to problems in fluid mechanics
65N08 Finite volume methods for boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage

Cite

References:

[1]Edwards, M. G.; Rogers, C. F., Finite volume discretization with imposed flux continuity for the general tensor pressure equation, Comput. Geom., 2, 259-290 (1998) ·Zbl 0945.76049
[3]Edwards, M. G.; Pal, M., Positive definite q-families of continuous subcell darcy-flux cvd(mpfa) finite volume schemes and the mixed finite element method, Internat. J. Numer. Methods Fluids, 57, 355-387 (2008) ·Zbl 1236.76035
[5]Aavatsmark, I., Introduction to multipoint flux approximation for quadrilateral grids, Comput. Geom., 6, 405-432 (2002) ·Zbl 1094.76550
[6]Lee, S. H.; Jenny, P.; Tchelepi, H. A., A finite-volume method with hexahedral multiblock grids for modeling flow in porous media, Comput. Geom., 6, 353-379 (2002) ·Zbl 1094.76541
[7]Wheeler, M. F.; Yotov, I., A multipoint flux mixed finite element method, SIAM J. Numer. Anal., 44, 2082-2106 (2006) ·Zbl 1121.76040
[8]Edwards, M. G., Higher-resolution hyperbolic - coupled - elliptic flux-continuous cvd schemes on structured and unstructured grids in 3-d, Int. J. Numer. Methods Fluids, 51, 1079-1095 (2006) ·Zbl 1158.76364
[9]Raviart, R. A.; Thomas, J. M., A mixed finite element method for second order problems, (Mathematical Aspects of the Finite Element Method. Mathematical Aspects of the Finite Element Method, Lectures Notes in Math., vol. 606 (1977)), 292-315 ·Zbl 0362.65089
[10]Edwards, M. G.; Zheng, H., A quasi-positive family of continuous darcy-flux finite volume schemes with full pressure support, J. Comput. Phys., 227, 22, 9333-9364 (2008) ·Zbl 1231.76178
[12]Aavatsmark, I.; Eigestad, G. T.; Mallison, B. T.; Nordbotten, J. M., A compact multipoint flux approximation method with improved robustness, Numer. Methods Partial Diff. Eqns, 25, 5, 1329-1360 (2008) ·Zbl 1230.65114
[13]Le Potier, C., Schema volume finis pour des operator operateurs de diffusion fortement anisotropes sur des maillages de triangle nonstructures, C. R., Math. Acad. Sci. Paris, Ser. I., 340, 921-926 (2005) ·Zbl 1076.76049
[14]Lipnikov, K.; Shashkov, M.; Svyatskiy, D.; Vassilevski, Yu., Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes, J. Comput. Phys., 227, 492-512 (2008) ·Zbl 1130.65113
[15]Edwards, M. G., Unstructured, control-volume distributed, full-tensor finite volume schemes with flow based grids, Comput. Geom., 6, 3-4, 433-452 (2002) ·Zbl 1036.76034
[16]Edwards, M. G., \(M\)-matrix flux splitting for general full tensor discretization operators on structured and unstructured grids, J. Comput. Phys., 160, 1-28 (2000) ·Zbl 0983.76055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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