Partitioned-Matrix acceleration to the Fission-Source iteration of the Variational Nodal Method

The Variational Nodal Method (VNM) expands the nodal volumetric flux and surface partial current into the sums of orthogonal basis functions without using the transverse integration technique. The exclusion of the transverse integration provides a number of advantages for employing the VNM in Pressurized Water Reactor (PWR) core simulation. The orthogonality of those basis functions guarantees the conservation of neutron balance regardless of the expansion orders, providing an opportunity to accelerate the computationally expensive full-order iteration by using cheap low-order sweeping with high-order moments fixed. This was named as the Partitioned-Matrix (PM) technique in the legacy VNM code VARIANT, and was applied to the within-group (WG) iteration. It is very effective for neutron-transport calculation, but less effective for neutron-diffusion mainly due to the reduced number of high-order partial current moments. In this paper, we extend the PM technique to the Fission-Source (FS) iteration to accelerate the flux convergence by using low-order flux moments also. From the macroscopic acceleration point of view, it converges the fission- and scattering-source distributions by using computationally cheap low-order iteration faster than the original full-order sweeping. Based on our new VNM code VIOLET, considering the fact that the discontinuity factor used for preserving neutron leakage rates during spatial homogenization slows down the nodal iteration convergence, numerical tests were carried out for two typical PWR problems respectively without and with discontinuity factors. By analyzing both the computational effort in terms of FLOP (FLoating-point OPeration) and computing time, the following conclusions have been demonstrated. The legacy PM technique for WG iteration can provide an acceleration ratio of about 2 for the PWR core neutron-diffusion calculation with or without using discontinuity factors, while the one for FS iteration itself can accelerate by a factor of about 3 which is higher. By accelerating both the WG and FS iteration simultaneously, the acceleration ratio is about 4 for both the two PWR problems. In addition, by extending the PM technique from the WG iteration to the FS iteration, the neutron-diffusion calculation of the VNM can be accelerated very effectively with almost no extra storage or implementation cost to the existing computer code.