Development of a semi-analytical nodal methodology for one-dimensional eigenvalue problems based on multigroup neutron transport theory using the discrete ordinate formulation

A semi-analytical nodal methodology is described in this paper for obtaining the numerical solution of eigenvalue problems based on neutron transport theory, in slab geometry, with isotropic scattering, using the discrete ordinates and multigroup energy formulation. This new approach applies a quadratic polynomial approximation only to the fission term in the transport equation, which justifies the method being classified as semi-analytical. The numerical solution is obtained through two interconnected iterative processes: the outer iterative process, which uses the power method to obtain successive estimates for the fission source and the effective multiplication factor ( ), and the internal iterative process, which aims to obtain successive estimates of the angular neutron fluxes emerging from the homogeneous regions throughout the spatial domain. Once the eigenvalues are calculated, for each outer power iteration, the angular fluxes of neutrons emerging at the node faces and in the sweeping direction of the internal iterative process are estimated. These are incoming fluxes at the faces of adjacent nodes, which ensures the continuity of the numerical solution. Computational algorithms were implemented in MATLAB. Numerical results for a typical benchmark problem considering an ADS, c.f., Accelerator-Driven Subcritical Reactor, type model are provided to illustrate the accuracy of the converged numerical solutions in coarse-mesh calculations. Although the method is not free from spatial truncation error, the results for the benchmark problem were considered satisfactory with one node per region, and refining the spatial domain did not result in a high computational cost in terms of response time, with results approaching those of analytical solutions.