Deterministic Neutron Transport
Deterministic transport codes discretize the Boltzmann transport equation on grids in space, angle, and energy, complementing Monte-Carlo methods.
Solving the transport equation directly
Instead of sampling particle histories, deterministic codes solve the linear Boltzmann transport equation numerically. They discretize the neutron flux in space (a mesh), in direction (discrete ordinates, the Sn method), and in energy (multigroup bins), turning the integro-differential equation into a large linear system solved by iteration.
The discrete-ordinates method
In Sn methods the angular variable is represented by a finite set of directions with quadrature weights. The flux is swept across the mesh direction by direction, with scattering coupling the directions and energy groups through source iteration. Acceleration schemes speed the convergence of the iterations.
Strengths
- Smooth, low-noise flux everywhere, including deep in shields
- No statistical error, so weak fluxes are as well resolved as strong ones
- Efficient for problems dominated by attenuation over long paths
Limitations
Deterministic methods introduce discretization errors instead of statistical ones: ray effects from too few angles, multigroup errors from coarse energy bins, and mesh errors in complex geometry. Representing intricate three-dimensional engineering detail is harder than in Monte-Carlo, which handles arbitrary geometry naturally.
Hybrid methods
The two approaches are complementary and are often combined: a fast deterministic solve provides an importance map that guides variance reduction in a Monte-Carlo run. This hybrid gives Monte-Carlo fidelity in complex geometry with deterministic efficiency for deep penetration, a common strategy for demanding shielding problems.
In fusion neutronics, deterministic and Monte-Carlo results are compared as an independent check on shielding and flux predictions.