Nodal collocation method for the multidimensional PL equations applied to neutron transport source problems

Abstract

A P L spherical harmonics-nodal collocation method is applied to the solution of the multidimensional neutron source transport equation. Vacuum boundary conditions are approximated by setting Marshak's conditions. The method is applied to several 1D, 2D and 3D problems with isotropic fixed source and with isotropic and anisotropic scattering. These problems are chosen to test this method in limit conditions, showing that in some cases a high order P L approximation is required to obtain accurate results and convergence. Results are also compared with the ones provided by several reference codes showing good agreement. It is also shown that Marshak's approximation to vacuum boundary conditions gives the same results that simulating vacuum with a purely absorbing medium and setting zero flux boundary conditions.

Validation of the SHNC time-dependent transport code based on the spherical harmonics method for complex nuclear fuel assemblies

“…The time discretization method described in previous Section has been incorporated into the FORTRAN 90 code SHNC (Spherical Harmonics-Nodal Collocation), obtaining an extended version of the code, which we use to solve time-dependent transport problems. The first version of the SHNC code for stationary problems was developed by the authors in previous works and validated with criticality problems [36,6] and also with stationary internal and external source problems [37]. The code is able to solve the P L approximation to the transport equation, for arbitrary odd order L, for arbitrary number of energy groups, for multidimensional space in general rectangular geometries and with isotropic and anisotropic scattering and sources.…”

“…(1). To achieve a finite approximation, the expansions given by series (7) and ( 8) are truncated at some finite order l = L, i.e., φ lm = s lm = 0, for l > L (the so-called P L closure condition [35]) and the resulting equations are the P L equations, see for example [36,6,37] for a full development. In the following, we will only consider L to be an odd integer because, as a consequence of the interface conditions given by Eqs.…”

“…where (x 1 , x 2 , x 3 ) are Cartesian coordinates, Σ a = diag(Σ t − Σ sl ) l=even ,Σ a = diag(Σ t − Σ sl ) l=odd are square diagonal matrices, and M j andM j are rectangular matrices (of dimension n e × n o and n o ×n e , respectively) that are described in previous works [36,6,37]. It is worth noting that, due to the definite parity of the spherical harmonics, the even l and odd l moments are the coefficients of the even-parity and odd-parity neutronic flux Φ ± = 1 2 Φ( Ω) ± Φ(− Ω) = l= even odd l m=−l φ lm Y m l that appear in the derivation of the even-parity transport equation [38].…”

Numerical analysis of the 2D C5G7 MOX benchmark using PL equations and a nodal collocation method

“…Beyond this, an increase of the number of computer cores does not reduce the computational time, but has the advantage of increasing the amount of global memory available to the computer code. The code also solves the isotropic fixed source problem for an arbitrary P L approximation for odd order L (Capilla et al, 2016).…”

A study of the radiative transfer equation using a spherical harmonics-nodal collocation method

“…In this Section, we present some numerical results in order to evaluate the performance and numerical accuracy of the nodal collocation method. The method has been already tested with neutron transport problems that are driven by internal sources [31] and also with eigenvalue problems [29]. The formulation described above has been implemented in the multi-group radiation transport code SHNC (Spherical Harmonics-Nodal Collocation), written in FORTRAN 90, which solves the external fixed source problem for an arbitrary P L approximation for odd L, and we show its application as a light propagation model for biological homogeneous and heterogeneous tissues by choosing appropriate numerical examples.…”