Determination of SSOR-SI iteration parameters

Abstract

Several conjugate gradient-like (CG-like) methods are applied to solve the nonsymmetric linear systems of equations derived from the time-dependent two-dimensional two-energy-group neutron diffusion equations by a finite difference approximation. The CG-like methods are: the generalized conjugate residual method; the generalized conjugate gradient least-square method; the generalized minimal residual method (GMRES); the conjugate gradient square method; and a variant of bi-conjugate gradient method (Bi-CGSTAB). In order to accelerate these CG-like methods, several preconditioning techniques are investigated. Two of the preconditioners are based on pointwise incomplete factorization: the incomplete LU (ILU) and the modified incomplete LU (MILU) decompositions. Another two, based on the block tridiagonal structure of the coefficient matrix, are blockwise and modified blockwise incomplete factorizations, BILU and MBILU. Finally, the last two are the alternating-direction implicit and symmetric successive overrelaxation (SSOR) preconditioners, which are derived from the basic iterative schemes. Comparisons are made by using CG-like methods combined with different preconditioners to solve a sequence of time-step reactor transient problems. We also compare the numerical results of CG-like methods with that of the line SOR method and the hybrid SSOR-SI Krylov subspace method, which is the Chebyshev semi-iteration method with the preconditioner SSOR, and with the GMRES method to estimate the extreme eigenvalues for the acceleration parameters. Numerical tests indicate that preconditioned BI-CGSTAB with the preconditioner MBILU requires less CPU time and number of iterations than other methods in reactor kinetics computation. Another advantage shown is that the preconditioneed CG-like methods are less sensitive to the time-step size used than the Chebyshev semi-iteration method and line SOR method. Comparison between CG-like methods indicates that the CGS, Bi-CGSTAB and GMRES methods are, on average, better than the other methods in reactor kinetics computation. Moreover, numerical results indicate that a good preconditioner is more important than the choice of CG-like methods. Numerical results also show that the MILU decomposition based on the conventional row-sum criterion has difficulty yielding a convergent solution in reactor kinetics computation. An improved version of the MILU decomposition is introduced by using Stone's idea for reactor kinetics computation.