Evaluation of spatial differencing practices for the discrete-ordinates method

Three popular spatial differencing practices for the discrete ordinates method are examined in detail for a basic two-dimensional Cartesian coordinates problem. These differencing schemes are 1) positive, 2) step, and 3) diamond schemes. The diamond scheme is shown to produce negative intensities under certain conditions irrespective of the number of control volumes employed, requiring some form of negative intensity fix-up. In absorbing-emitting or absorbing-emitting-scattering media, grid refinement can result in negative intensities when the diamond scheme is used. The diamond scheme and a positive scheme, which sets the negative intensities encountered in the diamond scheme to zero or very small number for purely absorbing media, can also produce physically unrealistic overshoots. The step scheme, although not considered as accurate as the diamond scheme, gives physically realistic results for the basic problem considered. Further evaluation of Fiveland's positive conditions, and variable weight and exponential-type schemes indicate a need for alternate spatial differencing schemes that describe the physics of radiative heat transfer more accurately.