Analytic Representation of the Solution of Neutron Kinetic Transport Equation in Slab-Geometry Discrete Ordinates Formulation

Abstract

Described here is a semi-analytical numerical method for the solution of simplified monoenergetic SN kinetics equations in a homogeneous slab, assuming one group of delayed neutron precursors. The basic idea relies on the solution of the neutron SN kinetics equation following the idea of the Decomposition method. To this end, the neutron angular flux and the concentration of the delayed neutron precursors are expanding in a truncation series of unknown functions, \(\sum _{k=0}^{M}\psi _{m}^{(k)}(x,t)\) and \(\sum _{k=0}^{M}C^{(k)}(x,t)\). Replacing these expansions in the SN transport equation we come up with one equation, in which, a simple count indicates that the kinetic problem has been reduced to a set of two equations in (2N + 2) unknowns, namely the expansions modes ψ(k) and C(k) for k = 1: M. In order to determine these unknown functions, we construct a recursive scheme of SN kinetic equations with the property that all equations satisfy the resulting equation, with the error being governed by the order M of the expansion. We remark that the first equation of the recursive system satisfy the boundary conditions of the SN kinetic equation whereas the remaining equations satisfy the homogeneous boundary conditions. Further we note that the construction of the recursive system is not unique. We support our choice with the argument that the analytical representation for each equation attained is known and given by the Time Laplace Transform Sn approach, i.e., the TLTSN method. This method has been implemented for the solution of a homogeneous slab and the numerical results have been compared with the ones available in the literature. Also, a convergence analysis for the recursive system are to be discussed.