Radiative intensity is defined for any direction Ω, because the angular space is continuous. In order to handle the radiative intensity equation numerically, the angular space is discretized.
The discrete ordinates method provides a discretization of angular space. The quadrature sets Level Symmetric Even, Level Symmetric Hybrid and Equal Weight Odd are designed using moment-matching conditions (see Ref. 21) and yield S
N approximations with
N(N + 2) directions in 3D or
N(N + 2)/2 in 2D (see
Ref. 20). The quasi-uniform quadrature set yields T
N approximations with
8N2 directions in 3D or
4N2 in 2D (
Ref. 42,
Ref. 43). These approximations are not designed using moment-matching conditions but allow for higher-order discretization compared with S
N approximations.
Depending on the value of N, a set of
n dependent variables has to be defined and solved for
I1,
I2, …,
In.
where Si is the
i-th discrete ordinate, with the following boundary condition
