Discrete Ordinates Method Implementation in 2D
For a given index i, define two indices, i+ and i, so that
Ω, Si+, and Si- have the same components in the xy–plane
and Si+ and Si- have opposite components in the z direction.
Assuming that a model is invariant in the z direction, the radiative transfer equation in two directions, Si+ and Si-, for the discrete ordinates method (DOM) reads:
Halving the sum of the two equations above and using ( in 2D) yields
which may recast
or
In addition if ϕ(SiSj) can be rewritten as a function of Si ⋅ Sj, as it is in the COMSOL Multiphysics implementation, then
and
In addition
so the above equation can be simplified:
(4-127)
with
since the third component of is zero in 2D.
Also notice that
(4-128)
with .
Using results from Equation 4-127 and Equation 4-128 the DOM is formulated in 2D using only radiative intensities, , on half of the 3D DOM directions, , except for the scattering term. In other expressions than the scattering term, the z component of the radiative intensities Ii and of the discrete directions Ωi can be ignored (or set to zero) and the weight wi, multiplied by 2.