Application of two preconditioned generalizedconjugate gradient methods to three-dimensional neutron and photon transport equations

Abstract

In this paper the preconditioned generalized conjugate gradient methods are applied to solve the linear system of equations that arise in three-dimensional neutron and photon transport equations. These generalized conjugate gradient methods are the CGS (conjugate gradient square) algorithm and the Bi-CGSTAB (bi-conjugate gradient stabilized) algorithm. Several subroutines are developed from these preconditioned generalized conjugate gradient methods for use in time-independent multi-group three-dimensional neutron and photon transport equations. These subroutines are connected to the computer program TORT. The reason for choosing the preconditioned generalized conjugate gradient methods is that these methods have good residual error control procedures during computation and have good convergence rates. We test a problem having an 8 set of matrix equations with 24255 unknowns each, in a personal computer with an AMD Athlon-XP1600+ central processing unit (CPU) and using Mandrake Linux 6.3 as the operating system. The point-wise incomplete LU factorization (ILU) and modified point-wise incomplete LU factorization (MILU) are the preconditioning techniques used in the test problem. We find that the preconditioned CGS and Bi-CGSTAB methods with the preconditioner ILU are more efficient than with the preconditioner MILU in the test problem. The numerical solution of flux by the preconditioned Bi-CGSTAB and CGS methods produces the same results as those obtained bythe successive over relaxation method (SOR) in the TORT program which is usually used for the calculation of radiation shielding in nuclear power plant and nuclear spent fuel storage.