Analytic solutions and numerical method for a coupled thermo-neutronic problem

We consider in this contribution a simplified idealized one-dimensional model in a nuclear core reactor coupling the diffusion equation on the neutron flux with the enthalpy equation for the water which collects the heat produced by this idealized nuclear core. These equations are coupled through the dependency of the coefficients of the diffusion equation in terms of the enthalpy. We propose a numerical method treating globally the coupled problem for finding its unique solution. Simultaneously, we use incomplete elliptic integrals to represent analytically the density of neutrons and the enthalpy in the fluid. Both methods lead to the same solution with high accuracy. However, another quantity, generally used as a benchmark for comparing results, depends considerably on the approximation used for the coefficients of the diffusion equation.