Numerical methods and HPC
Open Access
Issue
Oil & Gas Science and Technology - Rev. IFP Energies nouvelles
Volume 73, 2018
Numerical methods and HPC
Article Number 82
Number of page(s) 17
DOI https://doi.org/10.2516/ogst/2018064
Published online 21 December 2018
  • Di Pietro D.A. (2012) Cell centered galerkin methods for diffusive problems, Math. Model. Numer. Anal. 46, 1, 111–144. [CrossRef] [EDP Sciences] [Google Scholar]
  • Aavatsmark I., Barkve T., Bøe Ø., Mannseth T. (1996) Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media, J. Comput. Phys. 127, 1, 2–14. [Google Scholar]
  • Agélas L., Di Pietro D., Droniou J. (2010) The G method for heterogeneous anisotropic diffusion on general meshes, M2AN Math. Model. Numer. Anal. 44, 4, 597–625. [CrossRef] [Google Scholar]
  • Agélas L., Guichard C., Masson R. (2010) Convergence of finite volume MPFA O type schemes for heterogeneous anisotropic diffusion problems on general meshes, Int. J. Finite Vol. 7, 2, 1–33. [Google Scholar]
  • Agélas L., Masson R. (2008) Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes, C. R. Math. 346, 17, 1007–1012. [CrossRef] [Google Scholar]
  • Edwards M., Rogers C. (1998) Finite volume discretization with imposed flux continuity for the general tensor pressure equation, Comput. Geosci. 2, 4, 259–290. [Google Scholar]
  • Wolff M., Cao Y., Flemisch B., Helmig R., Wohlmuth B. (2013) 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. [Google Scholar]
  • Eymard R., Gallouët T., Herbin R. (2010) Discretization of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: a scheme using stabilization and hybrid interfaces, IAM J. Num. Anal. 30, 4, 1009–1043. [CrossRef] [MathSciNet] [Google Scholar]
  • Eymard R., Herbin R., Guichard C., Masson R. (2012) Vertex-centred discretization of multiphase compositional Darcy flows on general meshes, Comput. Geosci. 16, 4, 987–1005. [Google Scholar]
  • Arnold D., Brezzi F. (1985) Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, Math. Model. Numer. Anal. 19, 1, 7–32. [CrossRef] [Google Scholar]
  • Raviart P.A., Thomas J.M. (1977) A mixed finite element method for 2-nd order elliptic problems, in: I. Galligani, E. Magenes (eds), Mathematical aspects of finite element methods, Springer, Berlin, Heidelberg, pp. 292–315. [CrossRef] [Google Scholar]
  • Brezzi F., Lipnikov K., Shashkov M. (2005) Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes, SIAM J. Numer. Anal. 43, 5, 1872–1896. [Google Scholar]
  • Brezzi F., Lipnikov K., Simoncini V. (2005) A family of mimetic finite difference methods on polygonal and polyhedral meshes, Math. Models Methods Appl. Sci. 15, 10, 1533–1551. [Google Scholar]
  • Droniou J., Eymard R., Gallouët T., Herbin R. (2010) A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods, Math. Models Methods Appl. Sci. 20, 2, 265–295. [Google Scholar]
  • Droniou J., Eymard R., Herbin R. (2016) Gradient schemes: generic tools for the numerical analysis of diffusion equations, ESAIM Math. Model. Numer. Anal. 50, 3, 749–781. [Google Scholar]
  • Vohralík M., Wohlmuth B.I. (2013) Mixed finite element methods: implementation with one unknown per element, local flux expressions, positivity, polygonal meshes, and relations to other methods, Math. Models Methods Appl. Sci. 23, 5, 803–838. [Google Scholar]
  • Danilov A., Vassilevski Y. (2009) A monotone nonlinear finite volume method for diffusion equations on conformal polyhedral meshes, Russ. J. Numer. Anal. Math. Modelling 24, 3, 207–227. [CrossRef] [Google Scholar]
  • Lipnikov K., Svyatskiy D., Vassilevski Y. (2009) Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes, J. Comput. Phys. 228, 3, 703–716. [Google Scholar]
  • Lipnikov K., Svyatskiy D., Vassilevski Y. (2012) Minimal stencil finite volume scheme with the discrete maximum principle, Russ. J. Numer. Anal. Math. Modelling 27, 4, 369–385. [CrossRef] [Google Scholar]
  • Potier C.L. (2005) Schéma volumes finis monotone pour des opérateurs de diffusion fortement anisotropes sur des maillages de triangles non structurés, C.R. Math. 341, 12, 787–792. [CrossRef] [MathSciNet] [Google Scholar]
  • Schneider M., Agélas L., Enchéry G., Flemisch B. (2017) Convergence of nonlinear finite volume schemes for heterogeneous anisotropic diffusion on general meshes, J. Comput. Phys. 351, Supplement C, 80–107. [Google Scholar]
  • Yuan G., Sheng Z. (2008) Monotone finite volume schemes for diffusion equations on polygonal meshes, J. Comput. Phys. 227, 12, 6288–6312. [Google Scholar]
  • Schneider M., Flemisch B., Helmig R. (2017) Monotone nonlinear finite-volume method for nonisothermal two-phase two-component flow in porous media, Int. J. Numer. Methods Fluids 84, 6, 352–381. [Google Scholar]
  • Schneider M., Flemisch B., Helmig R., Terekhov K., Tchelepi H. (Apr 2018) Monotone nonlinear finite-volume method for challenging grids, Comput. Geosci. 22, 2, 565–586. [Google Scholar]
  • Schneider M., Gläser D., Flemisch B., Helmig R. (2017) Nonlinear finite-volume scheme for complex flow processes on corner-point grids, in: C. Cancès, P. Omnes (eds), Finite volumes for complex applications VIII – Hyperbolic elliptic and parabolic problems, Springer International Publishing, pp. 417–425. [CrossRef] [Google Scholar]
  • Droniou J., Eymard R., Gallouët T., Guichard C., Herbin R. (2018) The gradient discretisation method, HAL, https://hal.archives-ouvertes.fr/hal-01382358. [Google Scholar]
  • Eymard R., Gallouët T., Herbin R. (2000) Finite volume methods, Handbook Numer. Anal. 7, 713–1018. [Google Scholar]
  • Evans L. (1998) Partial differential equations, Graduate Studies in Mathematics. American Mathematical Society. [Google Scholar]
  • Kuzmin D. (2010) A guide to numerical methods for transport equations, Universität Nürnberg, http://www.mathematik.uni-dortmund.de/~kuzmin/Transport.pdf. [Google Scholar]
  • Berman A., Plemmons R.J. (1994) Nonnegative matrices in the mathematical sciences, Vol. 9, SIAM, Philadelphia, PA. [CrossRef] [Google Scholar]
  • Droniou J., Potier C.L. (2011) Construction and convergence study of schemes preserving the elliptic local maximum principle, SIAM J. Numer. Anal. 49, 2, 459–490. [Google Scholar]
  • Potier C.L. (2009) A nonlinear finite volume scheme satisfying maximum and minimum principles for diffusion operators, Int. J. Finite Vol. 6, 2, 1–20. [Google Scholar]
  • Lipnikov K., Manzini G., Shashkov M. (2014) Mimetic finite difference method, J. Comput. Phys. 257, 1163–1227. [Google Scholar]
  • Di Pietro D.A., Vohralík M. (2014) A review of recent advances in discretization methods, a posteriori error analysis, and adaptive algorithms for numerical modeling in geosciences, Oil Gas Sci. Technol. - Rev. IFP Energies nouvelles 69, 4, 701–729. [CrossRef] [Google Scholar]
  • Droniou J. (2014) Finite volume schemes for diffusion equations: introduction to and review of modern methods, Math. Models Methods Appl. Sci. 24, 8, 1575–1619. [Google Scholar]
  • Agélas L., Eymard R., Herbin R. (2009) A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media, C. R. Math. 347, 11, 673–676. [CrossRef] [MathSciNet] [Google Scholar]
  • Aavatsmark I. (2002) An introduction to multipoint flux approximations for quadrilateral grids, Comput. Geosci. 6, 3–4, 405–432. [Google Scholar]
  • Hackbusch W. (1989) On first and second order box schemes, Computing 41, 4, 277–296. [CrossRef] [MathSciNet] [Google Scholar]
  • Helmig R. (1997) Multiphase flow and transport processes in the subsurface: a contribution to the modeling of hydrosystems, Springer-Verlag, Berlin. [Google Scholar]
  • Hommel J., Ackermann S., Beck M., Becker B., Class H., Fetzer T., Flemisch B., Gläser D., Grüninger C., Heck K., Kissinger A., Koch T., Schneider M., Seitz G., Weishaupt K. (2016) DuMuX 2.10.0, https://doi.org/10.5281/zenodo.159007. [Google Scholar]
  • Blatt M., Burchardt A., Dedner A., Engwer C., Fahlke J., Flemisch B., Gersbacher C., Gräser C., Gruber F., Grüninger C., Kempf D., Klöfkorn R., Malkmus T., Müthing S., Nolte M., Piatkowski M., Sander O. (2016) The distributed and unified numerics environment, version 2.4, Arc. Numer. Softw. 4, 100, 13–29. [Google Scholar]
  • Brezzi F., Lipnikov K., Shashkov M. (2006) Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes with curved faces, Math. Models Methods Appl. Sci. 16, 2, 275–297. [Google Scholar]
  • Fort J., Fürst J., Halama J., Herbin R., Hubert F. (2011) Finite Volumes for Complex Applications VI: Problems and Perspectives, Springer, Heidelberg. [CrossRef] [Google Scholar]
  • Krogstad S., Lie K., Møyner O., Nilsen H.M., Raynaud X., Skaflestad B. (2015) MRST-AD – an open-source framework for rapid prototyping and evaluation of reservoir simulation problems. SPE Reservoir Simulation Symposium, Houston, TX, February 23–25, Society of Petroleum Engineers. [Google Scholar]
  • Alkämper M., Dedner A., Klöfkorn R., Nolte M. (2016) The DUNE-ALUGrid module, Arc. Numer. Softw. 4, 1, 1–28. [Google Scholar]
  • Aarnes J.E., Krogstad S., Lie K.-A. (2008) Multiscale mixed/mimetic methods on corner-point grids, Comput. Geosci. 12, 3, 297–315. [Google Scholar]
  • Terekhov K., Mallison B., Tchelepi H. (2017) Cell-centered nonlinear finite-volume methods for the heterogeneous anisotropic diffusion problem, J. Comput. Phys. 330, 245–267. [Google Scholar]
  • Kapyrin I., Nikitin K., Terekhov K., Vassilevski Y. (2014) Nonlinear monotone FV schemes for radionuclide geomigration and multiphase flow models, in: Finite volumes for complex applications VII-Elliptic, parabolic and hyperbolic problems, Springer, pp. 655–663. [Google Scholar]
  • Aavatsmark I., Eigestad G., Mallison B., Nordbotten J. (2008) A compact multipoint flux approximation method with improved robustness, Numer. Methods Partial Differ. Equ. 24, 5, 1329–1360. [Google Scholar]
  • Scheck M., Bayer U. (1999) Evolution of the Northeast German Basin – inferences from a 3D structural model and subsidence analysis, Tectonophysics 3, 3, 145–169. [Google Scholar]
  • Woodside W., Messmer J. (1961) Thermal conductivity of porous media. I. Unconsolidated sands, J. Appl. Phys. 32, 9, 1688–1699. [Google Scholar]
  • Lipnikov K., Manzini G., Svyatskiy D. (2011) Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems, J. Comput. Phys. 230, 7, 2620–2642. [Google Scholar]

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