Treatment of Boundary Conditions in Trajectory-Based Deterministic Transport Methods

Boundary conditions are an essential part of the approximations used in the numerical solution of the transport equation. The collision probability and the characteristic methods are considered, and exact and approximated tracking methods to be used in the implementation of geometrical motions and albedo conditions are analyzed. The analysis of the exact boundary-condition treatment is carried out for finite domains and infinite lattices, where periodic trajectories must be used. Albedo-like boundary conditions may be used to approximate exact geometrical motions via spatially piecewise constant and either piecewise constant or discrete angular approximations for the boundary fluxes. We also have examined angular product quadrature formulas and shown that the recently proposed Bickley-Naylor quadratures do not respect particle conservation in the presence of anisotropy of scattering. Numerical examples show that the approximated albedo-type boundary method converges toward the results obtained with the exact boundary treatment. However, because of problems related to the multigroup implementation, numerical extra burden in group iterations prevents the efficient use of approximated boundary conditions for multigroup calculations. Nevertheless, this method remains a candidate of choice for use in multidomain calculations via interface boundary fluxes.