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An Adaptive Combined Preconditioner with Applications in Radiation Diffusion Equations

Part of: Partial differential equations, boundary value problems Numerical linear algebra

Published online by Cambridge University Press:  23 November 2015

Xiaoqiang Yue ,
Shi Shu ,
Xiao wen Xu  and
Zhiyang Zhou
Xiaoqiang Yue
Affiliation:
School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, P.R. China
Shi Shu*
Affiliation:
School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, P.R. China
Xiao wen Xu
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100088, P.R. China
Zhiyang Zhou
Affiliation:
School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, P.R. China
*
*Corresponding author. Email addresses: [email protected] (X. Yue), [email protected] (S. Shu), [email protected] (X. Xu), [email protected] (Z. Zhou)
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Abstract

The paper aims to develop an effective preconditioner and conduct the convergence analysis of the corresponding preconditioned GMRES for the solution of discrete problems originating from multi-group radiation diffusion equations. We firstly investigate the performances of the most widely used preconditioners (ILU(k) and AMG) and their combinations (Bco and Bco), and provide drawbacks on their feasibilities. Secondly, we reveal the underlying complementarity of ILU(k) and AMG by analyzing the features suitable for AMG using more detailed measurements on multiscale nature of matrices and the effect of ILU(k) on multiscale nature. Moreover, we present an adaptive combined preconditioner Bcoα involving an improved ILU(0) along with its convergence constraints. Numerical results demonstrate that Bcoα-GMRES holds the best robustness and efficiency. At last, we analyze the convergence of GMRES with combined preconditioning which not only provides a persuasive support for our proposed algorithms, but also updates the existing estimation theory on condition numbers of combined preconditioned systems.

Keywords

Radiation diffusion equationscombined preconditioneradaptive preconditioningconvergence analysis

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65F15: Eigenvalues, eigenvectors 65N55: Multigrid methods; domain decomposition 65Z05: Applications to physics
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 5 , November 2015 , pp. 1313 - 1335
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Pei, W. B., The construction of simulation algorithms for laser fusion, Commun. Comput. Phys. 2 (2) (2007), pp. 255–270.Google Scholar
[2]Fu, S. W., Fu, H. Q., Shen, L.J., Huang, S. K., and Chen, G. N., A nine point difference scheme and iteration solving method for two dimensional energy equations with three temperatures, Chin. J. Comput. Phys. 15 (4) (1998), pp. 489–497.Google Scholar
[3]Gu, T. X., Dai, Z. H., Hang, X. D., Fu, S. W., and Liu, X. P., Efficient algebraic methods for two-dimensional energy equations with three temperatures, Chin.J. Comput. Phys. 22 (6) (2005), pp. 471–478.Google Scholar
[4]Mo, Z. Y., Fu, S. W., and Shen, L.J., Parallel lagrange numerical simulations for 2-dimension three temperature hydrodynamics, Chin.J. Comput. Phys. 17 (6) (2000), pp. 625–632.Google Scholar
[5]Mo, Z. Y. and Fu, S. W., Application of krylov iterative methods in two dimensional three temperatures energy equation, J. Numer. Meth. Comput. Appl. 24 (2) (2003), pp. 133–143.Google Scholar
[6]Mo, Z. Y., Shen, L.J., and Wittum, G., Parallel adaptive multigrid algorithm for 2-D 3-T diffusion equations, Int. J. Comput. Math. 81 (3) (2004), pp. 361–374.Google Scholar
[7]Wu, J. P., Liu, X. P., Wang, Z. H., Dai, Z. H., and Li, X. M., Two preconditioning techniques for two-dimensional three-temperature energy equations, Chin. J. Comput. Phys. 22 (4) (2005), pp. 283–291.Google Scholar
[8]Xiao, Y. X., Shu, S., Zhang, P. W., Mo, Z. Y., and Xu, J. C., A kind of semi-coarsing AMG method for two dimensional energy equations with three temperatures, J. Numer. Meth. Comput. Appl. 24 (4) (2003), pp. 293–303.Google Scholar
[9]Xu, X. W., Mo, Z. Y., and An, H. B., Algebraic two-level iterative method for 2-D 3-T radiation diffusion equations, Chin. J. Comput. Phys. 26 (1) (2009), pp. 1–8.Google Scholar
[10]Baldwin, C., Brown, P. N., Falgout, R., Graziani, F., and Jones, J., Iterative linear solvers in 2D radiation-hydrodynamics code: methods and performance, J. Comput. Phys. 154 (1999), pp. 1–40.Google Scholar
[11]Saad, Y., Iterative Methods for Sparse Linear Systems, SIAM (2003).CrossRefGoogle Scholar
[12]Hysom, D. and Pothen, A., A scalable parallel algorithm for incomplete factor preconditioning, SIAM J. Sci. Comput. 22 (6) (2001), pp. 2194–2215.Google Scholar
[13]Brandt, A., McCormick, S. F., and Ruge, J., Algebraic multigrid (AMG) for automatic multi-grid solution with application to geodetic computations, Institute for Computational Studies (1982).Google Scholar
[14]Ruge, J. W. and Stüben, K., Algebraic multigrid, in Multigrid Methods, Front. Appl. Math. 3 (1987), pp. 73–130.Google Scholar
[15]Hu, X. Z., Wu, S. H., Wu, X. H., Xu, J. C., Zhang, C. S., Zhang, S. Q., and Zikatanov, L., Combined Preconditioning with Applications in Reservoir Simulation, Multiscale Model. Simul. 11 (2) (2013), pp. 507–521.Google Scholar
[16]Hang, X. D., Li, J. H., and Yuan, G. W., Convergence analysis on splitting iterative solution of multi-group radiation diffusion equations, Chin.J. Comput. Phys. 30 (1) (2013), pp. 111–119.Google Scholar
[17]Saad, Y. and Schultz, M. H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAMJ. Sci. Stat. Comput. 7 (3) (1986), pp. 856–869.CrossRefGoogle Scholar
[18]Henson, V. E. and Yang, U. M., BoomerAMG: a parallel algebraic multigrid solver and pre-conditioner, Appl. Numer. Math. 41 (2002), pp. 155–177.Google Scholar
[19]Xu, X. W., Research on scalable parallel algebraic multigrid algorithms, Ph.D. thesis (2007), Chinese Academy of Engineering Physics.Google Scholar
[20]Mo, Z. Y., Zhang, A. Q., Cao, X. L., Liu, Q. K., Xu, X. W., An, H. B., Pei, W. B., and Zhu, S. P., JASMIN: a parallel software infrastructure for scientific computing, Front. Comput. Sci. 4 (4) (2010), pp. 480–488.Google Scholar
[21]Peng, H. S., Zheng, Z.J., Zhang, B. H., et al., Direct-drive implosion experiments on the SG-II laser facility, J. Fusion Energ. 19 (1) (2000), pp. 81–85.Google Scholar
[22]Elman, H. C., Iterative methods for large sparse nonsymmetric systems of linear equations, Ph.D. thesis (1982), Yale University.Google Scholar