Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T23:03:41.692Z Has data issue: false hasContentIssue false
  • English
  • Français

A Lions Domain Decomposition Algorithm for Radiation Diffusion Equations on Non-matching Grids

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

Published online by Cambridge University Press:  10 November 2015

Li Yin ,
Jiming Wu  and
Zihuan Dai
Li Yin*
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100088, P. R. China
Jiming Wu
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100088, P. R. China
Zihuan Dai
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100088, P. R. China
*
*Corresponding author. Email address: yinli_kitty 122@sina.com (L. Yin), wu_jiming@iapcm.ac.cn (J. Wu), dai_zihuan@iapcm.ac. cn (Z. Dai)
Get access

Abstract

We develop a Lions domain decomposition algorithm based on a cell functional minimization scheme on non-matching multi-block grids for nonlinear radiation diffusion equations, which are described by the coupled radiation diffusion equations of electron, ion and photon temperatures. The L2 orthogonal projection is applied in the Robin transmission condition of non-matching surfaces. Numerical results show that the algorithm keeps the optimal accuracy on the whole computational domain, is robust enough on distorted meshes and curved surfaces, and the convergence rate does not depend on Robin coefficients. It is a practical and attractive algorithm in applying to the two-dimensional three-temperature energy equations of Z-pinch implosion simulation.

Keywords

Lions domain decompositionradiation diffusion equationsnon-matching gridsSchwarz algorithm

MSC classification

Secondary: 65M06: Finite difference methods 65M55: Multigrid methods; domain decomposition
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 4 , November 2015 , pp. 530 - 548
Copyright
Copyright © Global-Science Press 2015 

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]Aavatsmark, I., Barkve, T., Boe, O., Mannseth, T., Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media, J. Comput. Phys., vol.127, (1996), pp. 2–14.CrossRefGoogle Scholar
[2]Aavatsmark, I., Reiso, E., Teigland, R., Control-volume discretization method for quadrilateral grids with faults and local refinements, Comput. Geosci., (2001), pp. 1–23.CrossRefGoogle Scholar
[3]Achdou, Y., Japhet, C., Maday, Y., Nataf, F., A new cement to glue nonconforming grids with Robin interface conditions: the finite volume case, Numer. Math., vol.92, no.4, (2002), pp. 593–620.CrossRefGoogle Scholar
[4]Berndt, M., Lipnikov, K., Shashkov, M., Wheeler, M. F., Yotov, I., Superconvergence of the velocity in mimetic finite difference methods on quadrilaterals, SIAM J. Numer. Anal., vol.43, (2005), pp. 1728–1749.CrossRefGoogle Scholar
[5]Brezzi, F., Lipnikov, K., Simoncini, V., A family of mimetic finite difference methods on polygonal and polyhedral meshes, Math. Mod. Methods. Appl. Sci. vol.15, (2005), pp. 1533–1551.CrossRefGoogle Scholar
[6]Chen, Q. Y., Wan, J., Yang, Y., Mifflin, R. T., Enriched multi-point flux approximation for general grids, J. Comput. Phys., vol.227, (2008), pp. 1701–1721.CrossRefGoogle Scholar
[7]Ewing, R. E., Lazarov, R. D., Vassilevski, P. S., Local refinement techniques for elliptic problems on cell-centered grids, i: Error analysis. Math. Comput., vol.56, no.194, (1991), pp. 437–461.Google Scholar
[8]Fu, S., Fu, H., Shen, L., Huang, S., Chen, G., A nine point difference scheme and iteration solving method for two dimensional energy equations with three temperatures, Chinese J. Comput. Phys., vol.15, no.4, (1998), pp. 489–497.Google Scholar
[9]Gander, M. J., Optimized Schwarz methods, SIAM J. Numer. Anal. vol.44, no.2, (2006), pp. 699–731.CrossRefGoogle Scholar
[10]Gu, T., Dai, Z., Liu, X., Partial Newton-Krylov iterative methods for a system of energy equations with three-temperatures, Numerical Heat Transfer, Part B, vol.53, (2008), pp. 1–12.Google Scholar
[11]Hyman, J., Morel, J., Shashkov, M., Steinberg, S., Mimetic finite difference methods for diffusion equations, Comput. Geosci., vol.6 (2002), pp. 333–352.CrossRefGoogle Scholar
[12]Klausen, R. A., Winther, R., Robust convergence of multi point flux approximation on rough grids, Numer. Math., vol.104, (2006), pp. 317–337.CrossRefGoogle Scholar
[13]Knoll, D. A., Rider, W. J., Olson, G. L., An efficient nonlinear solution method for nonequilibrium radiation diffusion, J. Quant. Spectrosc. Radiat. Transfer, vol.63, (1999), pp. 15–29.CrossRefGoogle Scholar
[14]Lipnikov, K., Shashkov, M., Yotov, I., Local flux mimetic finite difference methods, Numer. Math., vol.112, (2009), pp. 115–152.CrossRefGoogle Scholar
[15]Lui, S. H., A Lions non-overlapping domain decomposition method for domains with an arbitrary interface, IMA J. Numer. Anal., vol.29, no.2, (2009), pp. 332–349.CrossRefGoogle Scholar
[16]Morel, J. E., Roberts, R. M., Shashkov, M. J., A local support-operators diffusion discretization scheme for quadrilateral r-z meshes, J. Comput. Phys., vol.144, (1998), pp. 17–51.CrossRefGoogle Scholar
[17]Mousseau, V. A., Knoll, D. A., Temporal accuracy of the nonequilibrium radiation diffusion equations applied to two-dimensional multimaterial simulations, Nuclear Sci. Engrg., vol.154, (2006), pp. 174–189.CrossRefGoogle Scholar
[18]Mousseau, V. A., Knoll, D. A., Rider, W. J., Physics-based preconditioning and the Newton-Krylov method for non-equilibrium radiation diffusion, J. Comput. Phys., vol.160, (2000), pp. 743–765.CrossRefGoogle Scholar
[19]Ober, C. C., Shadid, J. N., Studies on the accuracy of time-integration methods for the radiation-diffusion equations, J. Comput. Phys., vol.195, (2004), pp. 743–772.CrossRefGoogle Scholar
[20]Olson, G. L., Auer, L. H., Hall, M. L., Diffusion, P1, and other approximate forms of radiation transport, J. Quantum Spectrosc. Radiat. Transger, vol.64, (2000), pp. 619–634.CrossRefGoogle Scholar
[21]Saad, Y., Iterative method for sparse linear systems, New York :PWS publishing, 1996.Google Scholar
[22]Saas, L., Faille, I., Nataf, F., Willien, F., Finite volume methods for domain decomposition on nonmatching grids with arbitrary interface conditions, SIAM J. Numer. Anal., vol.43, (2005), pp. 860–890.CrossRefGoogle Scholar
[23]Sheng, Z., Yuan, G., A nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes, SIAM J. Sci. Comput., vol.30, (2008), pp. 1341–1361.CrossRefGoogle Scholar
[24]Sheng, Z., Yue, J., Yuan, G. , Monotone finite volume schemes for nonequilibrium radiation diffusion equations on distorted meshes, SIAM J. Sci. Comput., vol.31, (2009), pp. 2915–2934.CrossRefGoogle Scholar
[25]Wu, J., Dai, Z., Gao, Z., Yuan, G., Linearity preserving nine-point schemes for diffusion equation on distorted quadrilateral meshes, J. Comput. Phys., vol.229, (2010), pp. 3382–3401.CrossRefGoogle Scholar
[26]Yin, L., Wu, J., Gao, Z., The cell functional minimization scheme for diffusion problems on unstructured polygonal meshes, ESAIM: M2AN,vol.49,(2015), pp.193–220.CrossRefGoogle Scholar
[27]Yin, L., Wu, J., Yao, Y., A cell functional minimization scheme for parabolic problem, J. Comput. Phys., vol.229, (2010), pp. 8935–8951.CrossRefGoogle Scholar
[28]Yin, L., Wu, J., Yao, Y., A cell functional minimization scheme for domain decomposition method on non-orthogonal and non-matching meshes, Numer. Math., vol.128, (2014), pp.773–804.CrossRefGoogle Scholar
[29]Yuan, G., Sheng, Z., Monotone finite volume schemes for diffusion equations on polygonal meshes, J. Comput. Phys., vol.227, (2008), pp. 6288–6312.CrossRefGoogle Scholar
[30]Zhang, Y.et. al., Computational investigation of the magneto-Rayleigh-Taylor instability in Z-pinch implosions, Phys. Plasmas., vol.17, no.4, (2010), 042702.Google Scholar