# STABILITY ANALYSIS OF SOLUTIONS OBTAINED BY THE LTSN METHOD FOR A RADIATIVE TRANSFER PROBLEM IN A HIGHLY NON-HOMOGENEOUS MEDIUM

## Palavras-chave:

Radiative transfer equation, Laplace transform discrete ordinate method, Sparse direct solver## Resumo

Heat transfer can be a result of a radiative process. In such case, it is modeled by the ra-

diative transfer equation (RTE), which is an integro-differential mathematical model that simulates the

movement of photons in a medium. However, many other applications are also modeled with the RTE.

Here, the RTE outputs the radiance distribution in the considered medium given the boundary conditions,

source term, and inherent optical properties, such as the absorption and scattering coefficients as well as

the scattering phase function, in the case of a translucid medium. The domain is discretized into the az-

imuthal, polar and vertical dimensions. The azimuthal discretization is obtained by the finite expansion

of the Legendre polynomial of the scattering phase function, and by the radiance expansion using the

Fourier decomposition of cosines. The resulting number of azimuthal modes is equivalent to the order of

anisotropy of the medium. The discretization in polar direction domain is made by an approximation of

the corresponding integral in the RTE, and the number of polar angles denotes the employed quadrature

order. Consequently, the RTE is expressed by a set of linear differential equations, one for each azimuthal

mode. The selected case study tackles an anisotropic and non-homogeneous medium, where the vertical

domain is discretized in 80 regions, with 50 polar angles, and the number of azimuthal modes is 174.

Therefore, each azimuthal mode requires to solve a linear system of differential equations with 80x50,

or 4000 unknowns. The chosen solver is the LTSN method. It emerged in the early 1990s in the neutron

transport research, being further extended to solve radiative transfer problems. The aim of this work is

to optimize the number of azimuthal modes and of the quadrature nodes while obtaining stable solutions

with accurate values of radiance. Another approach that is presented here is to solve only one linear

system for all azimuthal modes, in a single step, with a much larger number of unknowns.