The Discrete Ordinates Method
Discrete ordinates solve the transport equation by discretizing angle into a fixed set of directions with quadrature weights.
Discretizing the angular variable
The discrete ordinates method, often written S_N, is the deterministic counterpart to Monte Carlo for radiation transport. Where Monte Carlo samples directions randomly, S_N chooses a fixed set of discrete angular directions (ordinates) with associated quadrature weights, and solves the transport equation along each direction. Space is discretized by finite differences, finite volumes, or finite elements, and energy by multigroup cross sections.
The result is a deterministic solution over the whole phase space at once, giving flux everywhere rather than statistical estimates at chosen tallies. This is valuable when the flux is needed throughout a domain, such as full-field shielding or activation maps.
Sweeps and iteration
For each direction and energy group, the discretized equations are solved by a transport sweep that marches through the mesh following the flow of particles. Because scattering couples directions and groups, the sweeps are wrapped in source iteration, often accelerated by diffusion synthetic acceleration to speed convergence in optically thick, scattering-dominated regions.
Ray effects and mitigation
The main artifact of S_N is ray effects: because only discrete directions are used, streaming from localized sources can produce unphysical striping in low-scattering regions. Increasing the angular order, using better quadrature sets, or applying first-collision source treatments reduces these effects.
- Angular quadrature sets (level-symmetric, Gauss) define ordinates
- Multigroup treatment of energy dependence
- Transport sweeps with source iteration and acceleration
- Ray effects from finite angular resolution
Discrete ordinates and Monte Carlo are complementary in neutronics: S_N gives full-field deterministic results efficiently for many geometries, while Monte Carlo handles complex three-dimensional detail exactly. Hybrid schemes use one to accelerate or bias the other.