KP1-Scheme for Acceleration of Upscatter Iterations over the Neutron Thermalization Region and the Fission Source in Solving a Subcritical Boundary Value Problem

Abstract

For the transport equation in three-dimensional \(r,\;\vartheta ,\;z\) geometry, a \(K{{P}_{1}}\)-scheme is constructed for accelerating the convergence of upscatter iterations over the neutron thermalization region and the fission source in solving a subcritical boundary value problem, consistent with the Weighted Diamond Differencing (WDD) scheme, and its generalization to the case of nodal Linear Discontinues (LD) and Linear Best (LB) schemes of the 3rd and 4th order of accuracy in spatial variables is considered. To solve the system for accelerating corrections, an algorithm based on the use of the cyclic splitting method was used, similar to that used earlier when constructing the \(K{{P}_{1}}\)-scheme for accelerating the convergence of inner iterations. An algorithm for determining the energy dependence for accelerating corrections of the \(K{{P}_{1}}\)-scheme for accelerating the convergence of upscatter iterations is considered. The choice of a criterion for the convergence of upscatter iterations is considered, and a criterion integral over up-scattered thermal neutrons for the convergence of upscatter iterations over the region of neutron thermalization is proposed. A modification of the algorithm for the case of three-dimensional \(x,\;y,\;z\) geometry is considered. Numerical examples of using the \(K{{P}_{1}}\)-scheme for accelerating the convergence of upscatter iterations to solve typical problems of neutron transport in three-dimensional geometry are given.