Discrete ordinates method

In the case of time-independent monochromatic radiation in an elastically scattering medium, the RTE is[1]

${\displaystyle \mathbf {s} \cdot \nabla I(\mathbf {r} ,\mathbf {s} )=\kappa I_{b}(\mathbf {r} )-\beta I(\mathbf {r} ,\mathbf {s} )+{\frac {\sigma }{4\pi }}\int _{4\pi }I(\mathbf {r} ,\mathbf {s} )\Phi (\mathbf {s} ,\mathbf {s'} )d\Omega '}$

where the first term on the RHS is the contribution of emission, the second term the contribution of absorption and the last term is the contribution from scattering in the medium. The variable ${\displaystyle \mathbf {s} }$ is a unit vector that specifies the direction of radiation and the variable ${\displaystyle \mathbf {s'} }$ is a dummy integration variable for the calculation of scattering from direction ${\displaystyle \mathbf {s'} }$ to direction ${\displaystyle \mathbf {s} }$.

In the discrete ordinates method, the full solid angle of ${\displaystyle 4\pi }$ is divided to some number of discrete angular intervals, and the continuous direction variable ${\displaystyle \mathbf {s} }$ is replaced by a discrete set of direction vectors ${\displaystyle \mathbf {s_{j}} }$. Then the scattering integral in the RTE, which makes the solution problematic, becomes a sum[1][2]

${\displaystyle \sum _{j=1}^{n}w_{j}I(\mathbf {r} ,\mathbf {s_{j}} )\Phi (\mathbf {s_{j}} ,\mathbf {s_{i}} )}$

where the numbers ${\displaystyle w_{j}}$ are weighting coefficients for the different direction vectors. With this the RTE becomes a linear system of equations for a multi-index object, the number of indices depending on the dimensionality and symmetry properties of the problem.

It is possible to solve the resulting linear system directly with Gauss–Jordan elimination,[2] but this is problematic due to the large memory requirement for storing the matrix of the linear system. Another way is to use iterative methods, where the required number of iterations for a given degree of accuracy depends on the strength of scattering.[4][5]

The discrete ordinates method, or some variation of it, is applied for solving radiation intensities in several physics and engineering simulation programs, such as COMSOL Multiphysics[6] or the Fire Dynamics Simulator.[7]

• Radiative transfer
• Thermal radiation
• Neutron radiation
• Bickley-Naylor functions