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A Spherical-Harmonics Solution for Radiative-Transfer

Problems with Reflecting Boundaries and Internal


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,

91509-900 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 spherical-harmonics method, including some recent improve-

ments, is used to establish the complete solution for a general prob-

lem concerning radiative transfer in a plane-parallel 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 non-normally 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 spherical-harmonics method to solve this class of


We let I(o; ?; ') denote the intensity (radiance) of the radiation field and

utilize the equation of transfer [2] for a plane-parallel medium for our model.

We write

? @

@o I(o; ?; ') + I(o; ?; ') = $


Z 1


Z 2ss

p(cos ?)I(o; ?0; '0)d'0d?0 + S0(o)


Preprint submitted to Elsevier Preprint 16 October 1996