On the use of the conjugate gradient method for the solution of the neutron transport equation

The possibility of using the standard conjugate gradient (CG) method to directly solve the Sn-equations based on the diamond difference scheme is studied for mono-energetic neutron transport problems with isotropic scattering. It is shown that such a direct use is possible for practical heterogeneous problems with a significant speed-up over the conventional source iteration (SI) method except for the problems that are prone to unphysical negative fluxes. Some recipes are suggested to make use of the CG-method even in those cases which need negative flux fix-up in the SI-method. The transport synthetic acceleration scheme, recently developed by Ramone [Nucl. Sci. Eng. 125 (1997) 257] and others, is shown to be useful in such cases. A symmetrisation scheme for the coefficient matrix has also been presented to enable the use of the CG-method. This scheme is compared with another approach of using weighted inner products.