discreteordinates

The discrete ordinates method, commonly abbreviated SN, is a numerical technique used to solve linear transport equations for particles such as neutrons or photons. It discretizes the angular dependence of the particle flux into a finite set of directions, turning a continuous integral over angles into a weighted sum. This angular discretization is then coupled with spatial discretization to produce a solvable system of equations.

In SN, the unit direction set {omega_m} and corresponding weights {w_m} are chosen from a quadrature rule. The continuous angular flux psi(x, omega) is approximated by psi_m(x) = psi(x, omega_m) for each discrete direction m. For a one-dimensional or multi-dimensional geometry, the governing transport equation for each direction m takes the form mu_m d/ds psi_m(s) + sigma_t psi_m(s) = sum_{m'} sigma_s(m' -> m) psi_{m'}(s) + q_m(s), where mu_m is the component of omega_m along the principal axis, sigma_t is the total cross section, sigma_s represents scattering, and q_m is a source term for that direction. The right-hand side approximates the angular integral of the scattering source by a discrete sum over the directions.

Boundary conditions specify the incoming flux for directions entering the domain. The result is a coupled set of first-order differential equations in space, one for each discrete direction, which are solved with standard spatial discretization methods (finite difference, finite element, or discontinuous Galerkin). The scalar flux is then obtained by phi(x) = sum_m w_m psi_m(x).

SN is widely used in nuclear reactor physics, radiation shielding, and astrophysical radiative transfer. It offers accurate solutions without diffusion approximations but can be memory-intensive and susceptible to ray effects, which are mitigated by higher-order or hybrid angular discretizations.