Comparison of the operation of the equations of the surface harmonics method and the finite difference method in the test problem

Abstract

Currently, one of the main methods of neutron-physical calculation of the reactor is the homogenization method, in which after obtaining effective small-group characteristics of cells, the heterogeneous core, in fact, turns into a piecewise homogeneous one. To find the distribution of neutrons in such a zone, the diffusion equation is solved by finite-difference (or nodal) methods. One of the methods justifying this approach is the surface harmonics method (SHM), which in the initial period of its development acted as a justification and refinement of the homogenization method. In the simplest versions of the SHM, the resulting finite-difference equations are reduced to a form similar to the finite-difference approximation of the diffusion equation. It is interesting to compare in the simplest cases the advantages and disadvantages of a simple finite-difference approximation of the diffusion equation and the finite-difference equations obtained using SHM. An analytical comparison is made using the example of two-dimensional geometry. To do this, the paper briefly describes how to obtain equations in SHM.