
A SphericalHarmonics Solution for RadiativeTransfer
Problems with Reflecting Boundaries and Internal
Sources
L. B. Barichello a, R. D. M. Garcia b and C. E. Siewert c
a Instituto de Matem?atica, Universidade Federal do Rio Grande do Sul,
91509900 Porto Alegre, RS, Brazil
b Centro T?ecnico Aeroespacial, Instituto de Estudos Avan?cados,
12231{970 S~ao Jos?e dos Campos, SP, Brazil
c Center for Research in Scientific Computation, Mathematics Department,
North Carolina State University, Raleigh, NC 27695{8205, USA
The sphericalharmonics method, including some recent improve
ments, is used to establish the complete solution for a general prob
lem concerning radiative transfer in a planeparallel medium. An
L  th order Legendre polynomial expansion of the phase function
is allowed, internal sources and reflecting boundaries are included
in the model, and since a nonnormally incident beam is impinging
on one surface, all components in a Fourier decomposition of the
intensity are required in the solution. Numerical results for two test
problems are reported.
1 Introduction
Some years ago, the FN method was used [1] to solve a general radiative
transfer problem that was based on a model that included internal sources,
reflecting boundaries and a beam incident on one surface. Here we use some
recent improvements in the sphericalharmonics method to solve this class of
problems.
We let I(o; ?; ') denote the intensity (radiance) of the radiation field and
utilize the equation of transfer [2] for a planeparallel medium for our model.
We write
? @
@o I(o; ?; ') + I(o; ?; ') = $
4ss
Z 1
?1
Z 2ss
p(cos ?)I(o; ?0; '0)d'0d?0 + S0(o)
(1)
Preprint submitted to Elsevier Preprint 16 October 1996