The discrete ordinates approximation replaces the continuous direction variable of the linear Boltzmann (transport / radiative transfer) equation with a finite set of ordinates and a quadrature rule, reducing an integro-differential equation to a system of first-order differential equations. The Analytical Discrete Ordinates formulation solves that system by seeking exponential solutions, producing an eigenvalue problem of half the number of discrete directions whose eigenvectors give the angular flux written explicitly in the spatial variable, with an arbitrary half-range quadrature in place of Gauss-Legendre and a scaling of the exponentials that prevents overflow. A student learns how to derive that eigensystem, how transverse integration over rectangular nodes extends it to two-dimensional Cartesian geometry, and how the choice of angular quadrature controls truncation error and the ray effect.
The linear Boltzmann equation - called the transport equation for neutrons and the radiative transfer equation for photons - balances particle gains and losses in phase space, coupling a streaming te…