what does SN and PN abrevations means in the discrete ordinates method used to solve the RTE ?

In discrete ordinates methods, SN refers to a level symmetric quadrature where the number of ordinates (think directions) is N(N+2).

Imagine a unit sphere center about three local Cartesian axes zeta, eta, xi. We divide each axis zeta, eta, xi into N equal segments (zeta1, zeta2, ... and have zeta_1 = eta_1 = xi_1, etc...) or for a single octant, we would have N/2 values for zeta, eta, xi (for an octant we go from 0 to 1 or 0 to -1 instead of -1 to 1) For N=8 (or N/2 = 4 for a single octant), we would get the following ordinate pattern on one octant of a unit sphere (below is just a sketch where the dashes "-" are used to make the text formatting look right when i post the message):

-------*

-----*----*

---*----*---*

-*---*----*---*

Of course, each octant has the same pattern and does not change for 90 degree rotations about the local Cartesian axes zeta, eta, xi.

As for PN, this usually refers to the Legnedre polynomial and in radiation heat transfer, PN methods approximate the directional dependence of the radiative intensity by an order N Legendre polynomial and take moments of the radiative transfer equation with respect to solid angle to create a closed set of equations.

In 1-D cases, a good quadrature for DOM is Guassian divisions and weights, which is actually derived from the zeros of Legendre Polynomials, so this may have been where you saw PN being used in relation to DOM.

thanks Jeff, for your clarifications. in fact I'm interested to the finite volume method to resolve the RTE. seeing that it has some similarities with DOM, that's why I've seen few articles on this Method. so have you some idees on the FVM for radiations? if it's so, have you seen the reference " Prediction of radiative transfer in cylindrical enclosures with the finite volume method" written by Chui & Raithby 1992 Journal of thermodynamics and heat transfer ? I've some questions on this article.