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 quasiminimal residual variant of the biconjugate gradient stabilized algorithm (QMRCGSTAB) (Chan et ai., 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 TFQM Rand QM RCGST AB 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 (BiCGST AB). In order to accelerate the TFQM Rand QM RCGST AB 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 MBlLU 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 QMRCGST AB methods in three typical reactor kinetics test problems. Numerical experiments indicate that preconditioned QMRCGST AB 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, TFQM R, QM RCGST AB, CGS,