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A Finite Volume Scheme for Three-Dimensional Diffusion Equations

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

Published online by Cambridge University Press:  14 September 2015

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 Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, P.R. China
Guangwei Yuan*
Affiliation:
Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, P.R. China
*
*Corresponding author. Email addresses: qxlai2000@sdu.edu.cn (X. Lai), sheng_zhiqiang@iapcm.ac.cn (Z. Sheng), yuan_guangwei@iapcm.ac.cn (G. Yuan)
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Abstract

The extension of diamond scheme for diffusion equation to three dimensions is presented. The discrete normal flux is constructed by a linear combination of the directional flux along the line connecting cell-centers and the tangent flux along the cell-faces. In addition, it treats material discontinuities by a new iterative method. The stability and first-order convergence of the method is proved on distorted meshes. The numerical results illustrate that the method appears to be approximate second-order accuracy for solution.

Keywords

Finite volume schemediffusion equationtetrahedral meshes

MSC classification

Secondary: 65M08: Finite volume methods 65M12: Stability and convergence of numerical methods 65M55: Multigrid methods; domain decomposition

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 3 , September 2015 , pp. 650 - 672
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Li, D., Shui, H. and Tang, M., On the finite difference scheme of two-dimensional parabolic equation on non-rectangular mesh, Numer, J.. Methods Comput. Appl., 1 (1980), pp. 217–224.Google Scholar
[2]Huang, W. and Kappen, A.M., A study of cell-center finite volume methods for diffusion equations, Mathematics Research Report, 98-10-01, University of Kansas, Lawrence KS66045.Google Scholar
[3]Hermeline, F., A finite volume method for the approximation of diffusion operators on distorted meshes, J. Comput. Phys., 160 (2000), pp. 481–499.CrossRefGoogle Scholar
[4]Hermeline, F., Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes, Comput. Methods Appl. Mech. Eng., 192 (2003) 1939–1959.Google Scholar
[5]Yuan, G. and Sheng, Z., Analysis of accuracy of a finite volume scheme for diffusion equations on distorted meshes, J. Comput. Phys., 224 (2007), pp. 1170–1189.CrossRefGoogle Scholar
[6]Morel, J.E., Dendy, J.E., Hall, M.L. and White, S.W., A cell centered Lagrangian-mesh diffusion differencing scheme, J. Comput. Phys., 103 (1992) 286–299.Google Scholar
[7]Shashkov, M. and Steinberg, S., Support-operator finite-difference algorithms for general elliptic problems, J. Comput. Phys., 118 (1995) 131–151.Google Scholar
[8]Shashkov, M. and Steinberg, S., Solving diffusion equations with rough coefficients in rough grid, J. Comput. Phys., 129 (1996) 383–405.CrossRefGoogle Scholar
[9]Hyman, J., Shashkov, M. and Steinberg, S., The numerical solution of diffusion problems in strongly heterogeneous non-isotropic material, J. Comput. Phys., 132 (1997) 130–148.Google Scholar
[10]Aavatsmark, I., An introduction to multipoint flux approxmations for quadrilateral grids, Comput. Geosci., 6 (2002), 405–432.Google Scholar
[11]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 (1998) 1700–1716.Google Scholar
[12]Aavatsmark, I., Barkve, T., Boe, O., Mannseth, T., Discretization on unstructured grids for inhomogeneous, anisotropic media Part II: Discussion and numerical results, SIAM J. Sci. Comput., 19 (1998) 1717–1736.CrossRefGoogle Scholar
[13]Chen, G., Li, D., Wan, Z., Difference scheme based on variation principle for two dimensional heat conduction equation, Chinese Journal of Computational Physics, 19 (2002) 299–304.Google Scholar
[14]Sheng, Z. and Yuan, G., A nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes, SIAM J. Sci. Comput., 30 (2008) 1341–1361.CrossRefGoogle Scholar
[15]Yuan, G. and Sheng, Z., Calculating the vertex unknowns of nine point scheme on quadrilateral meshes for diffusion equation, Science in China Series A: Mathematics, 51 (2008) 1522–1536.Google Scholar
[16]Edwards, M.G. and Zheng, H., A quasi-positive family of continuous Darcy-flux finite-volume schemes with full pressure support, J. Comput. Phys., 227 (2008) 9333–9364.Google Scholar
[17]Friis, H.A. and Edwards, M.G.. A family of MPFA finite-volume schemes with full pressure support for the general tensor pressure equation on cell-centered triangular grids, J. Comput. Phys., 230 (2011) 205–231.Google Scholar
[18]Lipnikov, K., Shashkov, M. and Yotov, I., Local flux mimetic finite difference methods, Technical Report LA-UR-05-8364, Los Alamos National Laboratory, 2005.Google Scholar
[19]Lipnikov, K., Morel, J., Shashkov, M., Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes, J. Comp. Phys. 199 (2004) 589–597.Google Scholar
[20]Chen, G., Li, D., Wan, Z., Difference scheme by integral interpolation method for three dimensional diffusion equation, Chinese Journal of Computational Physics, 20 (2003), 205–209.Google Scholar
[21]Du, Z., Yin, D., Liu, X., Lu, J., Nonorthogonal hexahedralmesh finite volume difference method for the 3-D diffusion equation, J. Tsinghua Univ (Sci & Tech), 43 (2003) 1365–1368.Google Scholar
[22]Yin, D., Du, Z., Lu, J., An unstructured tetrahedral mesh finite volume difference method for the diffusion equation, J. Numer. Methods Comput. Appl., 2 (2005) 92–100.Google Scholar
[23]Coudiére, Y., Pierre, C., Rousseau, O., and Turpault, R.. A 2d/3d discrete duality finite volume scheme. Application to ecg simulation, Int. Journal on Finite Volumes, (2009), 6(1).Google Scholar
[24]Coudiére, Y., Hubert, F., A 3d discrete duality finite volume method for nonlinear elliptic equation. HAL Preprint URL http://bal.archives-ouvertes.fr/docs/00/45/68/37/PDF/ddfv3d.pdf.Google Scholar
[25]Coudiére, Y., Hubert, F., Manzini, G., Benchmark 3D:CeVeFE-DDFV, a discrete duality scheme with cell/vertex/face+edge unknowns, Fořt, J. et al, Finite volumes for complex applications VI-problems & perspectives, Springer proceedings in mathematics 4, DOI 10.1007/978-3-642-20671-9-95, ©Springer-Verlag Berlin Heidelberg 2011.Google Scholar
[26]Hermeline, F., Approximation of 2-D and 3-D diffusion operators with variable full tensor coefficients on arbitrary meshes, Comput. Method Appl. Mech. Eng., 196 (2007) 2497–2526.Google Scholar
[27]Hermeline, F., A finite volume method for approximating 3D diffusion operators on general meshes, Journal of Computational Physics, 228 (2009) 5763–5786.Google Scholar
[28]Edwards, M.G., Zheng, H., 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 (2011) 455–487.Google Scholar
[29]Lipnikov, K., Shashkov, M., Svyatskiy, D., The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes, J. Comput. Phys., 211 (2006) 473–491.CrossRefGoogle Scholar
[30]Wu, J., Dai, Z., Gao, Z., Yuan, G., Linearity preserving nine-point schemes for diffusion equation on distorted quadrilateral meshes, J. Comput. Phys., 229 (2010) 3382–3401.CrossRefGoogle Scholar
[31]Bessemoulin-Chatard, M., Chainais-Hillairet, C. and Filbet, F., On discrete functional inequalities for some finite volume schemes, 2012, arXiv:1202.4860v1.Google Scholar
[32]Eymard, R., Gallouet, Th., and Herbin, R., Finite volume methods, In Handbook of numerical analysis, volume VII, 2000, pages 713–1020, North-Holland, Amsterdam.Google Scholar