Let’s define a cylindrical coordinate system (er, eΨ, ez) and the direction Ω which coordinates (μ, η, ξ) in this coordinate system are expressed with its azimuthal angle ϕ with respect to the basis vector er, and polar angle θ:

The radiative transfer equation has an additional term to account for the angular redistribution of the radiative intensity (see Ref. 23):

For the discrete ordinates method, the angular redistribution term is approximated by a finite differences central scheme (see Ref. 23):

In this equation, Ii-1/2 and Ii+1/2 are the intensities evaluated at the boundaries between the ordinates, and αi-1/2 and αi+1/2 are the angular redistribution coefficients. In turn, the intensities between ordinates are evaluated with a numerical scheme:

The angular redistribution term is calculated with the equations above, and allows the discrete ordinate method to solve for 2D axisymmetric models. The handling of the other terms of the radiative transfer equation is similar to 2D, noting that the symmetry plane is the rz-plane. Therefore, as in 2D, only half of the 3D directions need to be solved for.