Application of preconditioned transpose-free quasi-minimal residual method for two-group reactor kinetics

Two preconditioned transpose-free quasi-minimal residual methods (TFQMR) (Freund, SIAM J. Sci. Stat. Comput. 14, 470 1993) and quasi-minimal residual variant of the biconjugate gradient stabilized algorithm (QMRCGSTAB) (Chan et al., SIAM J. Sci. Stat. Comput. IS. 338 1994) are applied to solve the non-symmetric linear systems of equations which are derived from the time dependent two-dimensional two-energy-group neutron 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 gradient 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 pointwise incomplete factorization: the incomplete factorization (ILU) and the modified incomplete factorization (MILU). Another two based on the block tridiagonal structure of the coefficient matrix are blockwise and modified blockwise incomplete factorizations, BILU and MBILU which 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 overrelaxation (BSOR) preconditioners which are derived from the basic iterative schemes. Comparisons are made by these methods combined with different preconditioners to solve a sequence of time steps reactor transient problems. Numerical results indicate that the preconditioner significantly affects the convergent rate TFQMR and QMRCGSTAB methods in three typical reactor kinetics test problems. Numerical experiments indicate that preconditioned QMRCGSTAB with the preconditioner MBILU requires fewer iterations than other methods in the three typical reactor kinetics test problems. Moreover, numerical results indicate that a good preconditioner can significantly improve the total iteration number (i.e, rate of convergence) of these generalized conjugate gradient methods, TFQMR, QMRCGSTAB, CGS, Bi-CGSTAB and GMRES. For preconditioners and MBILU and BILU, we find that all of the eigenvalues of preconditioned matrix are more clustered about 1 than the eigenvalues of other preconditioners in a typical reactor kinetics test problem. Such eigenvalue distribution is very favorable or the rate of convergence of these generalized conjugate gradient methods. Copyright (C) 1996 Elsevier Science Ltd