Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T14:03:22.607Z Has data issue: false hasContentIssue false
  • English
  • Français

A Conservative Parallel Iteration Scheme for Nonlinear Diffusion Equations on Unstructured Meshes

Part of: Numerical methods in complex analysis Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  02 November 2016

Yunlong Yu ,
Yanzhong Yao ,
Guangwei Yuan  and
Xingding Chen
Yunlong Yu*
Affiliation:
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, P.R. China
Yanzhong Yao*
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
Xingding Chen*
Affiliation:
Department of Mathematics, College of Science, Beijing Technology and Business University, Beijing 100048, P.R. China
*
*Corresponding author. Email addresses:yu_yunlong@iapcm.ac.cn (Y. Yu), yao_yanzhong@iapcm.ac.cn (Y. Yao), yuan_guangwei@iapcm.ac.cn (G. Yuan), chenxd@th.btbu.edu.cn (X. Chen)
*Corresponding author. Email addresses:yu_yunlong@iapcm.ac.cn (Y. Yu), yao_yanzhong@iapcm.ac.cn (Y. Yao), yuan_guangwei@iapcm.ac.cn (G. Yuan), chenxd@th.btbu.edu.cn (X. Chen)
*Corresponding author. Email addresses:yu_yunlong@iapcm.ac.cn (Y. Yu), yao_yanzhong@iapcm.ac.cn (Y. Yao), yuan_guangwei@iapcm.ac.cn (G. Yuan), chenxd@th.btbu.edu.cn (X. Chen)
*Corresponding author. Email addresses:yu_yunlong@iapcm.ac.cn (Y. Yu), yao_yanzhong@iapcm.ac.cn (Y. Yao), yuan_guangwei@iapcm.ac.cn (G. Yuan), chenxd@th.btbu.edu.cn (X. Chen)
Get access

Abstract

In this paper, a conservative parallel iteration scheme is constructed to solve nonlinear diffusion equations on unstructured polygonal meshes. The design is based on two main ingredients: the first is that the parallelized domain decomposition is embedded into the nonlinear iteration; the second is that prediction and correction steps are applied at subdomain interfaces in the parallelized domain decomposition method. A new prediction approach is proposed to obtain an efficient conservative parallel finite volume scheme. The numerical experiments show that our parallel scheme is second-order accurate, unconditionally stable, conservative and has linear parallel speed-up.

Keywords

Diffusion equationsconservationparallel schemeunstructured mesh

MSC classification

Secondary: 65M08: Finite volume methods 65E05: Numerical methods in complex analysis (potential theory, etc.)
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 5 , November 2016 , pp. 1405 - 1423
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Dawson, C.N., Du, Q., Dupont, T.F., A finite difference domain decomposition algorithm for numerical solution of the heat equation, Math. Comput., 57(1991), 63–71.Google Scholar
[2] Dawson, C.N., Dupont, T.F., Explicit/implicit conservative Galerkin domain decomposition procedures for parabolic problems, Math. Comput., 58(1992), 21–34.Google Scholar
[3] Dawson, C.N., Dupont, T.F., Explicit/implicit conservative domain decomposition procedures for parabolic problems based on block-centered finite differences, SIAM J. Numer. Anal., 31(1994), 1045–1061.Google Scholar
[4] Dryja, M., Substructreing methods for parabolic problems, Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, PA, (1991), 264–271.Google Scholar
[5] Gao, Z.M., Wu, J.M, A linearity-preserving cell-centered scheme for the heterogeneous and anisotropic diffusion equations on general meshes, Int. J. Numer. Meth. Fluids, 67(2011), 2157–2183.Google Scholar
[6] Laevaky, Yu.M., Rudenko, O.V., Splitting methods for parabolic problems in nonrectangular domains, Appl.Math.Lett., 8:6(1995), 9–14.CrossRefGoogle Scholar
[7] Liao, H.L., Shi, H.S., Sun, Z.Z., Corrected explicit-implicit domain decomposition algorithms for two-dimensional semilinear parabolic equations, Sci. China Ser. A, 52:8 (2009), 2362–2388.Google Scholar
[8] Saad, Y., Iterative Method for Sparse Linear Systems, PWS Pub. Co., New York, 1996.Google Scholar
[9] Sheng, Z.Q., Yuan, G.W., A nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes, SIAM J. Sci. Comput., 30 (2008), 1341–1361.CrossRefGoogle Scholar
[10] Sheng, Z.Q., Yuan, G.W., Hang, X. D., Unconditional stability of parallel difference schemes with second order accuracy for parabolic equation, Appl. Math. Comput., 184 (2007), 1015–1031.Google Scholar
[11] Shi, H.S., Liao, H.L., Unconditional stability of corrected explicit-implicit domain decomposition algortihtms for parrallel approximation of heat equations, SIAM J. Numer. Anal., 44 (2006), 1584–1611.Google Scholar
[12] Toselli, A., Widlund, O., Domain Decomposition Methods-Algorithms and Theory, Springer-Verlag, Berlin, 2005.Google Scholar
[13] J.M Wu, Dai, Z.H., Gao, Z.M., Yuan, G.W., Linearity preserving nine-point schemes for diffusion equation on distorted quadrilateral meshes, J. Comput. Phys., 229(2010), 3382–3401.Google Scholar
[14] Yuan, G.W., Zuo, F.L., Parallel differences schemes for heat conduction equation, Int. J. Comput. Math., 80:8 (2003), 995–999.Google Scholar
[15] Yuan, G.W., Shen, L.J., Stability and convergence of explicit-implicit conservative domain decomposition procedure for parabolic problems, Appl. Math. Comput., 47(2004), 793–801.Google Scholar
[16] Yuan, G.W., Hang, X.D., Sheng, Z.Q., The unconditional stability of parallel difference schemes with second order convergence for nonlinear parabolic system, J. Partial Diff. Eqs., Vol.20, No.1(2007), 1–22.Google Scholar
[17] Yuan, G.W., Yao, Y.Z., Yin, L., A conservative domain decomposition procedure for nonlinear diffusion problems on arbitrary quadrilateral grids, SIAM J. Sci. Comput., 33:3 (2011), 1352–1368.Google Scholar
[18] Zhu, S.H., Conservative domain decomposition procedure with unconditional stability and second-order accuracy, Appl. Math. Comput., 216 (2010), 3275–3282.Google Scholar