An Adaptive Combined Preconditioner with Applications in Radiation Diffusion Equations

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.

[1]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[2]  Xiaolin Cao,et al.  JASMIN: a parallel software infrastructure for scientific computing , 2010, Frontiers of Computer Science in China.

[3]  V. E. Henson,et al.  BoomerAMG: a parallel algebraic multigrid solver and preconditioner , 2002 .

[4]  Alex Pothen,et al.  A Scalable Parallel Algorithm for Incomplete Factor Preconditioning , 2000, SIAM J. Sci. Comput..

[5]  Gabriel Wittum,et al.  Parallel adaptive multigrid algorithm for 2-d 3-t diffusion equations , 2004, Int. J. Comput. Math..

[6]  Xiantu He,et al.  Direct-Drive Implosion Experiments on the SG-II Laser Facility , 2000 .

[7]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[8]  J. W. Ruge,et al.  4. Algebraic Multigrid , 1987 .

[9]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[10]  Mo Ze.  PARALLEL LAGRANGE NUMERICAL SIMULATIONS FOR 2-DIMENSION THREE TEMPERATURE HYDRODYNAMICS , 2000 .

[11]  Ludmil T. Zikatanov,et al.  Combined Preconditioning with Applications in Reservoir Simulation , 2012, Multiscale Model. Simul..

[12]  H. Elman.  Iterative methods for large, sparse, nonsymmetric systems of linear equations , 1982 .

[13]  An Hengbin.  Algebraic Two-level Iterative Method for 2-D 3-T Radiation Diffusion Equations , 2009 .

[14]  Robert D. Falgout,et al.  Iterative Linear Solvers in a 2D Radiation-Hydrodynamics Code , 1999 .

[15]  Yuan Guangwei.  Convergence Analysis on Splitting Iterative Solution of Multi-group Radiation Diffusion Equations , 2013 .

[16]  Wenbing Pei.  The Construction of Simulation Algorithms for Laser Fusion , 2007 .