The Method of Spherical Harmonics ( PN -Approximation)

For a gray medium (or on a spectral basis) with known temperature distribution (or for the case of radiative equilibrium), the general problem of radiative transfer entails determining the radiative intensity from an integro-differential equation in five independent variables—three space coordinates and two direction coordinates—a prohibitive task. The method of spherical harmonics provides a vehicle to obtain an approximate solution of arbitrarily high order (i.e., accuracy), by transforming the equation of transfer into a set of simultaneous partial differential equations (PDEs). The approach was first proposed by Jeans [1] in his work on radiative transfer in stars. Further description of the method may be found in the books by Kourganoff [2], Davison [3], and Murray [4] (the latter two dealing with the closely related neutron transport theory). The spherical harmonics method is identical to the moment method described in Chapter 15, except that moments are taken in such a way as to take advantage of the orthogonality of spherical harmonics.

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