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Monotone Finite Volume Scheme for Three Dimensional Diffusion Equation on Tetrahedral Meshes

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  05 December 2016

Xiang Lai ,
Zhiqiang Sheng  and
Guangwei Yuan
Xiang Lai*
Affiliation:
Department of Mathematics, Shandong University, Jinan 250100, P.R. China
Zhiqiang Sheng*
Affiliation:
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, P.R.China
Guangwei Yuan*
Affiliation:
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, P.R.China
*
*Corresponding author. Email addresses:[email protected] (X. Lai), [email protected] (Z. Sheng), [email protected] (G. Yuan)
*Corresponding author. Email addresses:[email protected] (X. Lai), [email protected] (Z. Sheng), [email protected] (G. Yuan)
*Corresponding author. Email addresses:[email protected] (X. Lai), [email protected] (Z. Sheng), [email protected] (G. Yuan)
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Abstract

We construct a nonlinear monotone finite volume scheme for three-dimensional diffusion equation on tetrahedral meshes. Since it is crucial important to eliminate the vertex unknowns in the construction of the scheme, we present a new efficient eliminating method. The scheme has only cell-centered unknowns and can deal with discontinuous or tensor diffusion coefficient problems on distorted meshes rigorously. The numerical results illustrate that the resulting scheme can preserve positivity on distorted tetrahedral meshes, and also show that our scheme appears to be approximate second-order accuracy for solution.

Keywords

Monotonicityfinite volume schemediffusion equationtetrahedral meshes

MSC classification

Secondary: 65M06: Finite difference methods 65M12: Stability and convergence of numerical methods 65M55: Multigrid methods; domain decomposition
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 1 , January 2017 , pp. 162 - 181
Copyright
Copyright © Global-Science Press 2016 

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References

[1] 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 (2007).CrossRefGoogle Scholar
[2] Aavatsmark, I., An introduction to multipoint flux approxmations for quadrilateral grids, Comput. Geosci. 6, 405–432 (2002).CrossRefGoogle Scholar
[3] Aavatsmark, I., Barkve, T., Boe, O., Mannseth, T., Discretization on unstructured grids for inhomogeneous, anisotropic media Part I: Derivation of the methods. SIAM J. Sci. Comput. 19, 1700–1716 (1998).CrossRefGoogle Scholar
[4] Sheng, Zhiqiang and Yuan, Guangwei, A nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes. SIAM J. Sci. Comput. 30, 1341–1361 (2008).CrossRefGoogle Scholar
[5] Chang, L., Yuan, G., Cell-centered finite volume methods with flexible stencils for diffusion equations on general nonconforming meshes, Comput. Methods Appl. Mech. Engrg. 198, 1638–1646 (2009).CrossRefGoogle Scholar
[6] Wu, Jingming, Dai, Zihuan, Gao, Zhiming, Yuan, Guangwei, Linearity preserving nine-point schemes for diffusion equation on distorted quadrilateral meshes, J. Comput. Phys. 229, 3382–3401 (2010).CrossRefGoogle Scholar
[7] Korotov, S., Krizek, M., Neittaanmi, P., Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle, Math.Comput. 70, 107–119 (2000).CrossRefGoogle Scholar
[8] Mishev, I.D., Finite volume methods on Voronoi meshes, Numer. Methods PDEs. 12, 193–212 (1998).3.0.CO;2-J>CrossRefGoogle Scholar
[9] Keilegarlen, Eirik, Nordbotten, Jan M., Ivar Aavatsmark, Sufficient criteria are necessary for monotone control volume methods, Appl. Math. Lett. 22, 1178–1180 (2009).CrossRefGoogle Scholar
[10] Nordbotten, J. M., Aavatsmark, I., Eigestad, G. T., Monotonicity of control volume methods, Numer. Math. 106, 255–288 (2007).CrossRefGoogle Scholar
[11] Keilegarlen, E., Aavatsmark, I., Monotonicity for MPFA methods on triangular grids, Comput. Geosci. 15, 3–16 (2011).CrossRefGoogle Scholar
[12] Edwards, Michael G., Zheng, Hongwen, Double-families of quasi-positive Darcy-flux approximations with highly anisotropic tensors on structured and unstructured grids, J. Comput. Phys. 229, 594–625 (2010).CrossRefGoogle Scholar
[13] Edwards, Michael G. and Zheng, Hongwen, A quasi-positive family of continuous Darcy-flux finite-volume schemes with full pressure support. J. Comput. Phys. 227, 9333–9364 (2008).Google Scholar
[14] Lipnikov, K., Manzini, G., and Svyatskiy, D., Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems, J. Comput. Phys. 230, 2620–2642 (2011).CrossRefGoogle Scholar
[15] Lipnikov, K., Manzini, G., and Svyatskiy, D., Monotonocity conditions in the mimetic finite difference method. Fořt, J. et al., Finite volumes for complex applications VI-problems & perspectives, Springer proceedings in mathematics 4, DOI 10.1007/978-3-642-20671-9_69, © Springer-Verlag Berlin Heidelberg 2011.Google Scholar
[16] Sharma, P. and Hammett, G.W., Preserving monotonicity in anisotropic diffusion, J. Comput. Phys. 227, 123–142 (2007).CrossRefGoogle Scholar
[17] Liska, R. and Shashkov, M., Enforcing the discrete maximum principle for linear finite element solutions of second-Order elliptic problems, Commun. Comput. Phys. 3, 852–877 (2008).Google Scholar
[18] 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).CrossRefGoogle Scholar
[19] Yuan, Guangwei, Sheng, Zhiqiang, Monotone finite volume schemes for diffusion equations on polygonal meshes, J. Comput. Phys. 227, 6288–6312 (2008).CrossRefGoogle Scholar
[20] Sheng, Zhiqiang, Yuan, Guangwei, The finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes, J. Comput. Phys. 230, 2588–2604 (2011).Google Scholar
[21] Lipnikov, K., Svyatskiy, D., Vassilevski, Y., Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes, J. Comput. Phys. 228, 703–716 (2009)CrossRefGoogle Scholar
[22] Lipnikov, K., Svyatskiy, D., Vassilevski, Y., A monotone finite volume method for advection–diffusion equations on unstructured polygonal meshes, J. Comput. Phys. 229, 4017–4032 (2010).CrossRefGoogle Scholar
[23] Palmer, T. S., A point-centered diffusion differencing for unstructured meshes in 3-D, UCRL-JC-118730, 1994.CrossRefGoogle Scholar
[24] Chen, Guangnan, Li, Deyuan, Wan, Zhengsu, Difference scheme by integral interpolation method for three dimensional diffusion equation. Chinese Journal of Computational Physics, 20, 205–209 (2003).Google Scholar
[25] Du, Zhengping, Yin, Dongsheng, Liu, Xiaoyu, Lu, Jinfu, Nonorthogonal hexahedralmesh finite volume difference method for the 3-D diffusion equation. J Tsinghua Univ (Sci & Tech), 43, 1365–1368 (2003).Google Scholar
[26] Yin, Dongsheng, Du, Zhengping, Lu, Jinfu, An unstructured tetrahedral mesh finite volume difference method for the diffusion equation. J. Numer. Methods Comput. Appl. 2, 92–100 (2005).Google Scholar
[27] Lipnikov, K., Shashkov, M., Svyatskiy, D., The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes, J. Comput. Phys. 211, 473–491 (2006).Google Scholar
[28] Lai, Xiang, Sheng, Zhiqiang and Yuan, Guangwei, A Finite Volume Scheme for Three-Dimensional Diffusion Equations, Communications in Computational Physics, 18, 650–672 (2015).CrossRefGoogle Scholar
[29] Berman, A., Plemmons, R. J., Nonnegative matrices in the mathematical science, Academic Press, New York, 1979.Google Scholar
[30] Edwards, Michael G., Zheng, Hongwen, Quasi M-Matrix multifamily continuous Darcy-flux approximations with full pressure support on structured and unstructured grids in three dimension, SIAM J. Sci. Comput. 33, 455–487 (2011).CrossRefGoogle Scholar
[31] Pal, M., Edwards, M., Continuous Darcy-flux approximation for general 3-D grids of any element type, paper SPE 106486. In: SPE Reservior simulation symposium. Houston, Texas, USA, 26-28 February 2007.Google Scholar
[32] Kapyrin, I. V., A family of monotone methods for the numerical solution of three-dimensional diffusion problems on unstructured tetrahedral meshes, Mathematics. 416, 588–593 (2007).Google Scholar
[33] Danilov, A., Vassilevski, Yu.. A monotone nonlinear finite volume method for diffusion equations on conformal polyhedral meshes, Russ. J. Numer. Anal. Math. Modeling, 24, 207–227 (2009).Google Scholar
[34] Nikitin, K., Vassilevski, Yu., A monotone nonlinear finite volume method for advection-diffusion equations on unstructured polyhedral meshes in 3D, Russ. J. Numer. Anal. Math. Modelling, 25, 335–358 (2010).Google Scholar