Skip to main content

JASMIN-based Two-dimensional Adaptive Combined Preconditioner for Radiation Diffusion Equations in Inertial Fusion Research

  • Xiaoqiang Yue (a1), Xiaowen Xu (a2) and Shi Shu (a1)

We present a JASMIN-based two-dimensional parallel implementation of an adaptive combined preconditioner for the solution of linear problems arising in the finite volume discretisation of one-group and multi-group radiation diffusion equations. We first propose the attribute of patch-correlation for cells of a two-dimensional monolayer piecewise rectangular structured grid without any suspensions based on the patch hierarchy of JASMIN, classify and reorder these cells via their attributes, and derive the conversion of cell-permutations. Using two cell-permutations, we then construct some parallel incomplete LU factorisation and substitution algorithms, to provide our parallel -GMRES solver with the help of the default BoomerAMG in the HYPRE library. Numerical results demonstrate that our proposed parallel incomplete LU preconditioner (ILU) is of higher efficiency than the counterpart in the Euclid library, and that the proposed parallel -GMRES solver is more robust and more efficient than the default BoomerAMG-GMRES solver.

Corresponding author
*Corresponding author. Email address: (S. Shu)
Hide All
[1] Pei W.B., The construction of simulation algorithms for laser fusion, Commun. Comput. Phys. 2, 255270 (2007).
[2] Yue X.Q., Shu S., Xu X.W. and Zhou Z.Y., An adaptive combined preconditioner with applications in radiation diffusion equations, Commun. Comput. Phys. 18, 13131335 (2015).
[3] Pomraning G.C., The Equations of Radiation Hydrodynamics, Pergamon (1973).
[4] Haines B.M., Grinstein F.F. and Fincke J.R., Three-dimensional simulation strategy to determine the effects of turbulent mixing on inertial-confinement-fusion capsule performance, Phys. Rev. Lett. 89, 053302 (2014).
[5] 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, 140 (1999).
[6] Xiao Y.X., Shu S., Zhang P.W., Mo Z.Y. and Xu J., A kind of semi-coarsing AMG method for two dimensional energy equations with three temperatures, J. Numer. Meth. Comput. Appl. 24, 293303 (2003).
[7] Mo Z.Y., Parallel adaptive solution for two dimensional 3-T energy equation on UG, Comput. Visual Sci. 9, 165174 (2006).
[8] Jiang J., Huang Y., Shu S. and Zeng S., Some new discretiztion and adaptation and multigrid methods for 2-D 3-T diffusion equations, J. Comput. Phys. 224, 168181 (2007).
[9] Zhou Z.Y., Xu X.W., Shu S., Feng C.S. and Mo Z.Y., An adaptive two-level preconditioner for 2-D 3-T radiation diffusion equations, Chin. J. Comput. Phys. 29, 475483 (2012).
[10] Saad Y., Iterative Methods for Sparse Linear Systems, SIAM (2003).
[11] Hysom D. and Pothen A., A scalable parallel algorithm for incomplete factor preconditioning, SIAM J. Sci. Comput. 22, 21942215 (2001).
[12] Brandt A., Multi-level adaptive solutions to boundary value problems, Math. Comput. 31, 333390 (1977).
[13] Ruge J.W. and K. Stüben, Algebraic multigrid, in multigrid methods, Front. Appl. Math. 3, 73130 (1987).
[14] Zhou J., Hu X. Z., Zhong L.Q., Shu S. and Chen L., Two-grid methods for Maxwell eigenvalue problems, SIAM J. Numer. Anal. 52, 20272047 (2014).
[15] Xiao Y., Zhou Z. and Shu S., An efficient algebraic multigrid method for quadratic discretizations of linear elasticity problems on some typical anisotropic meshes in three dimensions, Numer. Linear Algebra Appl. 22, 465482 (2015).
[16] Hu Q.Y., Shu S. and Wang J.X., Nonoverlapping domain decomposition methods with a simple coarse space for elliptic problems, Math. Comput. 79 (272), 20592078 (2010).
[17] Li Y.H., Shu S., Xu Y.S., and Zou Q.S., Multilevel preconditioning for the finite volume method, Math. Comput. 81 (279), 13991428 (2012).
[18] Hu X.Z., Wu S.H., Wu X.H., Xu J., Zhang C.S., Zhang S.Q. and Zikatanov L., Combined pre-conditioning with applications in reservoir simulation, Multiscale Model. Simul. 11, 507521 (2013).
[19] 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, 480488 (2010).
[20] Cao X.L., Mo Z.Y., Liu X., Xu X.W., and Zhang A.Q., Parallel implementation of fast multipole method based on JASMIN, Sci. China. Inf. Sci. 54, 757766 (2011).
[21] Cheng T.P., Mo Z.Y. and Shao J.L., Accelerating groundwater flow simulation in MODFLOW using JASMIN-based parallel computing, Groundwater 52, 194205 (2014).
[22] Zhang A.Q., Mo Z.Y. and Yang Z., Three-level hierarchical software architecture for data-driven parallel computing with applications, J. Comput. Res. Dev. 51, 25382546 (2014).
[23] Xu X.W., Mo Z.Y., Liu Q.K. and An H.B., An implicit time-integration algorithm for diffusion equations with structured AMR and applications, Chin. J. Comput. Phys. 29, 684692 (2012).
[24] Shu S., Yue X.Q., Zhou Z.Y. and Xu X.W., Approximation and two-level algorithm of finite volume schemes for diffusion equations with structured AMR, Chin. J. Comput. Phys. 31, 390402 (2014).
[25] Henson V.E. and Yang U.M., BoomerAMG: A parallel algebraic multigrid solver and preconditioner, Appl. Numer. Math. 41, 155177 (2002).
[26] Berger M.J. and Oliger J., Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys. 53, 484512 (1984).
[27] Berger M.J. and Colella P., Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys. 82, 6484 (1989).
[28] Gibbs N.E., Poole W.G. and Stockmeyer P. K., An algorithm for reducing the bandwidth and profile of a sparse matrix, SIAM J. Numer. Anal. 13, 236250 (1976).
[29] George A. and Liu J.W.H., The evolution of the minimum degree ordering algorithm, SIAM Rev. 31, 119 (1989).
[30] George A., Nested dissection of a regular finite element mesh, SIAM J. Numer. Anal. 10, 345363 (1973).
[31] Saad Y. and Suchomel B., ARMS: An algebraic recursive multilevel solver for general sparse linear systems, Numer. Linear Algebra Appl. 9, 359378 (2002).
[32] Osei-Kuffuor D., Li R.P. and Saad Y., Matrix reordering using multilevel graph coarsening for ILU preconditioning, SIAM J. Sci. Comput. 37, A391A419 (2015).
[33] Saad Y., ILUT: A dual threshold incomplete ILU factorization, Numer. Linear Algebra Appl. 1, 387402 (1994).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 2
Total number of PDF views: 21 *
Loading metrics...

Abstract views

Total abstract views: 100 *
Loading metrics...

* Views captured on Cambridge Core between 7th September 2017 - 19th February 2018. This data will be updated every 24 hours.