A modified power method for the multilayer multigroup two-dimensional neutron diffusion equation

In this work we present a methodology of solution of the multigroup multi-layer stationary neutron diffusion equation in two-dimensional cartesian geometry. This eigenvalue problem describes the criticality of nuclear reactor, that is, it establishes the ratio between the numbers of neutrons generated in successive fission reactions. In order to solve this problem, we use the power method to obtain the dominant eigenvalue (Keff) and its corresponding eigenfunction. Each iteration of the power method requires the solution of a non-homogeneous diffusion problem, that usually is solved numerically, however in this work the neutron diffusion equation is solved in analytical form in each iteration. To solve this system of second order partial differential equations, we propose to use the Finite Fourier Transform in one of the spatial variables obtaining a transformed problem which is resolved by well-established methods for ordinary differential equations. After it is solved, we use the inverse Fourier Transform to reconstruct the expression of the neutron flux in the original variables. However, at each iteration of the power method it is necessary to update the source term with the neutron flux and the Keff of the previous iteration. Thus in all iterations new terms are added which becomes the process very laborious. To overcome this problem, the authors propose a methodology that approximates the neutron flux in standard form by polynomial interpolation. In order to reduce computational time we propose to subdivide the real regions of the problem into small fictitious regions. In this way, the interpolating polynomials of each region can be of low order, reducing the dimensions of the matrices involved and, consequently, computational time. The methodology is implemented to solve a heterogeneous problem and the numerical results are compared with the finite volumes method.