Solution for the Multigroup Neutron Space Kinetics Equations by Source Iterative Method

In this work, we used a modified Picard’s method to solve the Multigroup Neutron Space Kinetics Equations (MNSKE) in Cartesian geometry. The method consists in assuming an initial guess for the neutron flux and using it to calculate a fictitious source term in the MNSKE. A new source term is calculated applying its solution, and so on, iteratively, until a stop criterion is satisfied. For the solution of the fast and thermal neutron fluxes equations, the Laplace Transform technique is used in time variable resulting in a first order linear differential matrix equation, which are solved by classical methods in the literature. After each iteration, the scalar neutron flux and the delayed neutron precursors are reconstructed by polynomial interpolation. We obtain the fluxes and precursors through Numerical Inverse Laplace Transform by Stehfest method. We present numerical simulations and comparisons with available results in literature.


INTRODUCTION
The iterative source method has been applied with great success in the solution of the neutron transport equation (WILLERT, 2014) (ADAMS, 2012) (DANIELLA, 2017) and in the solution of the steady-state diffusion equation by (ZANETTE, 2017). In the last years many works have been developed in the search for solutions to the problem of kinetic neutron diffusion equation among them, it is possible to emphasize the works (NAHLA, 2012) (CEOLIN, 2011) (CORNO, 2008. In this context, the iterative source method has been used to solve the diffusion kinetic equation. The method consists of estimate an initial distribution for the source term of the fast flux equation decoupling the system and getting it possible to solve the equations separately. It should be noted that this occurs only in systems without up scattering. To solve the MNSKE, the authors apply the Laplace Transform, which transformed the set of partial differential equations into a set of ordinary differential equations (ODE's). These ODE's are solved from classical methods present in the literature. To return the functions to the frequency domain for time domain a numeric inversion of the Laplace was used for. After each iterative process, the term source is updated with the previous expressions of neutron fluxes and delayed neutron precursors. Due to the use of the numerical inverse transform, it is necessary to reconstruct the fluxes and concentrations through an interpolation. The authors prefer to use a polynomial interpolation in order to always maintain a standard structure for all iterations. This iterative process continues until a stop criterion is established.

TRON DIFFUSION THEORY
Starting from the Neutron Diffusion Spatial Kinetics equation for two energy groups, six groups of delayed neutron precursors, nuclear parameters not dependent on time and space, without external source and one-dimensional Cartesian geometry we have: with 1,..., 6 i = , subject to the following boundary conditions: where W represents the source term of the fast flux equation The constants 1  and 2  are obtained by replacing the boundary conditions given in (2).
Therefore, we have 1  and 1  equal to: Substituting the constants in the fast flux equation has the transformed equation written as: To solve (8) where 1 2 3 4 5 6 , , , , , y y y y y y are the coefficients of the interpolated polynomial.
where ( ) 2 s  is given by: The constants 3  and 4  are determined analogously to the constants 1  and 2  . Thus, the transformed thermal flux can be written as: Where ( ) ,0 i Cx are obtained as follows: where 1: 6. Therefore, we can solve the system of equations of (1) through an iterative process described as follows:

Stop criterion test
6. If the stop criterion is satisfied the process is terminated otherwise the index k is updated is returned to item 1.

RESULTS AND DISCUSSION
To validate the methodology proposed in this work we were used an algorithm implemented in the Scilab platform applied to a problem with two groups of energy and six groups of delayed neutron precursors with domain 0 160 x cm  , 0,8520306528 eff k = and nuclear parameters given by the tables 1and 2.  As the initial condition the solution was used for the steady-state diffusion equation solved by (ZANETTE, 2017). The Tables 3 and 4 show the numerical convergence of the fast and thermal flows in 1 = ts and 80 cm with the decrease of  , respectively. In the Table 3 to 4 equal to 5 10 − we have a 6-digit concordance comparing with 8 10 − , already for 7 10 − has an 8-digit agreement, which indicates that as  tends to zero the solution tends to stabilize. It can be observed in Table 4.

CONCLUSION
Analyzing the previous results, it can be observed that the methodology proposed in this article is promising and important in the solution of neutron space kinetics problems. Through the source iterative method, the system equations become decoupled which allows them were resolved separately becoming the solution of the problem simpler. Another important characteristic with respect to the methodology proposed is the achievement of a semi-analytic solution at each iteration. As future perspectives for the work, it is intended to apply the same methodology to the multi-region problem and to analyze the convergence of the method.

ACKNOWLEDGMENT
We would like to thank CAPES -Coordination of Superior Level Staff Improvement for supported this work.