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Monotone nonlinear finite-volume method for challenging grids. (English) Zbl 1405.65145

Summary: This article presents a new positivity-preserving finite-volume scheme with a nonlinear two-point flux approximation, which uses optimization techniques for the face stencil calculation. The gradient is reconstructed using harmonic averaging points with the constraint that the sum of the coefficients included in the face stencils must be positive. We compare the proposed scheme to a nonlinear two-point scheme available in literature and a few linear schemes. Using two test cases, taken from the FVCA6 benchmarks, the accuracy of the scheme is investigated. Furthermore, it is shown that the scheme is linearity-preserving on highly complex corner-point grids. Moreover, a two-phase flow problem on the Norne formation, a geological formation in the Norwegian Sea, is simulated. It is demonstrated that the proposed scheme is consistent in contrast to the linear Two-Point Flux Approximation scheme, which is industry standard for simulating subsurface flow on corner-point grids.

MSC:

65N08 Finite volume methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
86-08 Computational methods for problems pertaining to geophysics
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[1] Aarnes, J.E., Krogstad, S., Lie, K.A.: Multiscale mixed/mimetic methods on corner-point grids. Comput. Geosci. 12(3), 297-315 (2008) · Zbl 1259.76065 · doi:10.1007/s10596-007-9072-8
[2] Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6(3-4), 405-432 (2002) · Zbl 1094.76550 · doi:10.1023/A:1021291114475
[3] Aavatsmark, I.: Multipoint flux approximation methods for quadrilateral grids. In: 9th International Forum on Reservoir Simulation, Abu Dhabi (2007) · Zbl 1124.65101
[4] Aavatsmark, I.: Comparison of monotonicity for some multipoint flux approximation methods. R. Eymard et JM hérard (rédacteurs). Finite Volumes for Complex Applications, tome 5, 19-34 (2008) · Zbl 1422.65301
[5] Aavatsmark, I., Barkve, T., Bøe, Ø., Mannseth, T.: Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys. 127(1), 2-14 (1996) · Zbl 0859.76048 · doi:10.1006/jcph.1996.0154
[6] Agélas, L., Eymard, R., Herbin, R.: A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media. Comptes Rendus Mathématique 347(11), 673-676 (2009) · Zbl 1166.65051 · doi:10.1016/j.crma.2009.03.013
[7] Alkämper, M., Dedner, A., Klöfkorn, R., Nolte, M.: The DUNE-ALUGrid module. arXiv:1407.6954 (2014)
[8] Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO-Modé,lisation Mathématique et Analyse Numérique 19(1), 7-32 (1985) · Zbl 0567.65078
[9] Bertsimas, D., Tsitsiklis, J.N.: Introduction to Linear Optimization, vol. 6. MA, Athena Scientific Belmont (1997)
[10] Blatt, M., Burchardt, A., Dedner, A., Engwer, C., Fahlke, J., Flemisch, B., Gersbacher, C., Gräser, C., Gruber, F., Grüninger, C., et al.: The distributed and unified numerics environment, version 2.4. Archive of Numerical Software 4(100), 13-29 (2016)
[11] Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43(5), 1872-1896 (2005) · Zbl 1108.65102 · doi:10.1137/040613950
[12] Brezzi, F., Lipnikov, K., Simoncini, V.: A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15(10), 1533-1551 (2005) · Zbl 1083.65099 · doi:10.1142/S0218202505000832
[13] Cancès, C., Cathala, M., Le Potier, C.: Monotone corrections for generic cell-centered finite volume approximations of anisotropic diffusion equations. Numer. Math. 125(3), 387-417 (2013) · Zbl 1281.65139 · doi:10.1007/s00211-013-0545-5
[14] Cao, Y., Helmig, R., Wohlmuth, B.: The influence of the boundary discretization on the multipoint flux approximation l-method. Finite Volumes for Complex Applications V, ISTE, London, pp. 257-263 (2008) · Zbl 1422.65305
[15] Cao, Y., Helmig, R., Wohlmuth, B.: Geometrical interpretation of the multi-point flux approximation L-method. Int. J. Numer. Methods Fluids 60(11), 1173-1199 (2009) · Zbl 1166.76042 · doi:10.1002/fld.1926
[16] Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous Media, vol. 2. SIAM (2006) · Zbl 1092.76001
[17] Danilov, A., Vassilevski, Y.V.: A monotone nonlinear finite volume method for diffusion equations on conformal polyhedral meshes. Russ. J. Numer. Anal. Math. Model. 24(3), 207-227 (2009) · Zbl 1166.65394 · doi:10.1515/RJNAMM.2009.014
[18] Demmel, J.W., Eisenstat, S.C., Gilbert, J.R., Li, X.S., Liu, J.W.H.: A supernodal approach to sparse partial pivoting. SIAM J. Matrix Anal. Appl. 20(3), 720-755 (1999) · Zbl 0931.65022 · doi:10.1137/S0895479895291765
[19] Edwards, M.G., Rogers, C.F.: Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2(4), 259-290 (1998) · Zbl 0945.76049 · doi:10.1023/A:1011510505406
[20] Edwards, M.G., Zheng, H.: Quasi M-matrix multifamily continuous Darcy-flux approximations with full pressure support on structured and unstructured grids in three dimensions. SIAM J. Sci. Comput. 33(2), 455-487 (2011) · Zbl 1368.74070 · doi:10.1137/080745390
[21] Eymard, R., Guichard, C., Herbin, R.: Small-stencil 3d schemes for diffusive flows in porous media. ESAIM: Mathematical Modelling and Numerical Analysis 46(2), 265-290 (2012) · Zbl 1271.76324 · doi:10.1051/m2an/2011040
[22] Flemisch, B., Darcis, M., Erbertseder, K., Faigle, B., Lauser, A., Mosthaf, K., Müthing, S., Nuske, P., Tatomir, A., Wolff, M., et al.: DuMux: DUNE for multi-{phase, component, scale, physics,...} flow and transport in porous media. Adv. Water Resour. 34 (9), 1102-1112 (2011) · doi:10.1016/j.advwatres.2011.03.007
[23] Fort, J., Fürst, J., Halama, J., Herbin, R., Hubert, F.: Finite volumes for complex applications. VI. Problems & Perspectives, vol. 1, 2. Springer Proceedings in Mathematics (20011) · Zbl 1220.76004
[24] Friis, H.A., Edwards, M.G., Mykkeltveit, J.: Symmetric positive definite flux-continuous full-tensor finite-volume schemes on unstructured cell-centered triangular grids. SIAM J. Sci. Comput. 31(2), 1192-1220 (2008) · Zbl 1190.65163 · doi:10.1137/070692182
[25] Gao, Z., Wu, J.: A linearity-preserving cell-centered scheme for the heterogeneous and anisotropic diffusion equations on general meshes. Int. J. Numer. Methods Fluids 67(12), 2157-2183 (2011) · Zbl 1426.76367 · doi:10.1002/fld.2496
[26] Gao, Z., Wu, J.: A second-order positivity-preserving finite volume scheme for diffusion equations on general meshes. SIAM J. Sci. Comput. 37(1), A420-A438 (2015) · Zbl 1315.65077 · doi:10.1137/140972470
[27] Hoteit, H., Mosé, R., Philippe, B., Ackerer, P., Erhel, J.: The maximum principle violations of the mixed-hybrid finite-element method applied to diffusion equations. Int. J. Numer. Methods Eng. 55(12), 1373-1390 (2002) · Zbl 1062.76524 · doi:10.1002/nme.531
[28] Krogstad, S., Lie, K.A., Møyner, O., Nilsen, H.M., Raynaud, X., Skaflestad, B., et al.: Mrst-ad-an open-source framework for rapid prototyping and evaluation of reservoir simulation problems. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2015)
[29] Le Potier, C.: Schéma volumes finis monotone pour des opérateurs de diffusion fortement anisotropes sur des maillages de triangles non structurés. Comptes Rendus Mathématique 341(12), 787-792 (2005) · Zbl 1081.65086 · doi:10.1016/j.crma.2005.10.010
[30] Lipnikov, K., Manzini, G., Shashkov, M.: Mimetic finite difference method. J. Comput. Phys. 257, 1163-1227 (2014) · Zbl 1352.65420 · doi:10.1016/j.jcp.2013.07.031
[31] Lipnikov, K., Manzini, G., Svyatskiy, D.: Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems. J. Comput. Phys. 230(7), 2620-2642 (2011) · Zbl 1218.65117 · doi:10.1016/j.jcp.2010.12.039
[32] Lipnikov, K., Svyatskiy, D., Vassilevski, Y.: Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes. J. Comput. Phys. 228(3), 703-716 (2009) · Zbl 1158.65083 · doi:10.1016/j.jcp.2008.09.031
[33] Nikitin, K., Terekhov, K., Vassilevski, Y.: A monotone nonlinear finite volume method for diffusion equations and multiphase flows. Comput. Geosci. 18(3-4), 311-324 (2014) · Zbl 1378.76076 · doi:10.1007/s10596-013-9387-6
[34] Nordbotten, J., Aavatsmark, I., Eigestad, G.: Monotonicity of control volume methods. Numer. Math. 106(2), 255-288 (2007) · Zbl 1215.76090 · doi:10.1007/s00211-006-0060-z
[35] Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2-nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods, pp. 292-315. Springer (1977) · Zbl 0362.65089
[36] Schneider, M., Agélas, L., Enchéry, G., Flemisch, B.: Convergence of nonlinear finite volume schemes for heterogeneous anisotropic diffusion on general meshes. J. Comput. Phys. 351, 80-107 (2017). https://doi.org/10.1016/j.jcp.2017.09.003 · Zbl 1380.65340 · doi:10.1016/j.jcp.2017.09.003
[37] Schneider, M., Flemisch, B., Helmig, R.: Monotone nonlinear finite-volume method for nonisothermal two-phase two-component flow in porous media. Int. J. Numer. Methods Fluids 84(6), 352-381 (2016) · doi:10.1002/fld.4352
[38] Schneider, M., Gläser, D., Flemisch, B., Helmig, R.: Nonlinear finite-volume scheme for complex flow processes on corner-point grids. In: Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems: FVCA 8, Lille, France, June 2017, pp. 417-425. Springer International Publishing (2017) · Zbl 1365.76170
[39] Sheng, Z., Yuan, G.: The finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes. J. Comput. Phys. 230(7), 2588-2604 (2011) · Zbl 1218.65120 · doi:10.1016/j.jcp.2010.12.037
[40] Sun, W., Wu, J., Zhang, X.: A family of linearity-preserving schemes for anisotropic diffusion problems on arbitrary polyhedral grids. Comput. Methods Appl. Mech. Eng. 267, 418-433 (2013) · Zbl 1286.76138 · doi:10.1016/j.cma.2013.08.006
[41] Terekhov, K., Vassilevski, Y.: Two-phase water flooding simulations on dynamic adaptive octree grids with two-point nonlinear fluxes. Russ. J. Numer. Anal. Math. Model. 28(3), 267-288 (2013) · Zbl 1282.76126 · doi:10.1515/rnam-2013-0016
[42] Terekhov, K.M., Mallison, B.T., Tchelepi, H.A.: Cell-centered nonlinear finite-volume methods for the heterogeneous anisotropic diffusion problem. J. Comput. Phys. 330, 245-267 (2017) · Zbl 1380.65335 · doi:10.1016/j.jcp.2016.11.010
[43] Vidović, D., Dotlić, M., Dimkić, M., Pušić, M., Pokorni, B.: Convex combinations for diffusion schemes. J. Comput. Phys. 246, 11-27 (2013) · Zbl 1349.65584 · doi:10.1016/j.jcp.2013.03.034
[44] Wolff, M., Cao, Y., Flemisch, B., Helmig, R., Wohlmuth, B.: Multi-point flux approximation L-method in 3d: numerical convergence and application to two-phase flow through porous media. Radon Ser. Comput. Appl. Math., De Gruyter 12, 39-80 (2013) · Zbl 1302.76117
[45] Wu, J., Gao, Z.: Interpolation-based second-order monotone finite volume schemes for anisotropic diffusion equations on general grids. J. Comput. Phys. 275, 569-588 (2014) · Zbl 1349.65586 · doi:10.1016/j.jcp.2014.07.011
[46] Yuan, G., Sheng, Z.: Monotone finite volume schemes for diffusion equations on polygonal meshes. J. Comput. Phys. 227(12), 6288-6312 (2008) · Zbl 1147.65069 · doi:10.1016/j.jcp.2008.03.007
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