APPLICATION OF PRECONDITIONED TRANSPOSE-FREE QUASI-MINIMAL RESIDUAL METHOD FOR 2-GROUP REACTOR KINETICS

Two preconditioned transpose-free quasi-minimal residual methods (TFQM R) (Freund, SIAM J. Sci. Stat. Comput. 14, 470 1993) and quasi-minimal residual variant of the biconjugate gradient stabilized algorithm (QM RCGSTAB) (Chan et al., SIAM J. Sci. Stat. Comput. IS. 338 1994) are ap plied to solve the non-symmetric linear systems of equations which are derived from the time dependent two-dimensional two-energy-group neut ron diffusion equations by finite difference approximation. We compare the TFQMR and QMRCGSTAB methods with the other popular method such as the generalized minimal residual method (GMRES), the conjugate gradie nt square method (CGS), and biconjugate gradient stabilized algorithm (Bi-CGSTAB). In order to accelerate the TFQMR and QMRCGSTAB we use the preconditioning technique. Two of the preconditioners are based on po intwise incomplete factorization: the incomplete factorization (ILU) a nd the modified incomplete factorization (MILU). Another two based on the block tridiagonal structure of the coefficient matrix are blockwis e and modified blockwise incomplete factorizations, BILU and MBILU whi ch are suitable for the system of partial differential equations such as two-energy-group neutron diffusion equations. Finally, the last two are the alternating-direction implicit (ADI) and block successive ove rrelaxation (BSOR) preconditioners which are derived from the basic it erative schemes. Comparisons are made by these methods combined with d ifferent preconditioners to solve a sequence of time steps reactor tra nsient problems. Numerical results indicate that the preconditioner si gnificantly affects the convergent rate TFQMR and QMRCGSTAB methods in three typical reactor kinetics test problems. Numerical experiments i ndicate that preconditioned QMRCGSTAB with the preconditioner MBILU re quires fewer iterations than other methods in the three typical reacto r kinetics test problems. Moreover, numerical results indicate that a good preconditioner can significantly improve the total iteration numb er (i.e, rate of convergence) of these generalized conjugate gradient methods, TFQMR, QMRCGSTAB, CGS, Bi-CGSTAB and GMRES. For preconditione rs and MBILU and BILU, we find that all of the eigenvalues of precondi tioned matrix are more clustered about 1 than the eigenvalues of other preconditioners in a typical reactor kinetics test problem. Such eige nvalue distribution is very favorable or the rate of convergence of th ese generalized conjugate gradient methods. Copyright (C) 1996 Elsevie r Science Ltd