Least-Squares Spectral Element Method for Radiative Heat Transfer in Semitransparent Media

Chebyshev collocation spectral method for solving radiative transfer with the modified discrete ordinates formulations

“…However, it is difficult to obtain the exact analytical solution to the integro-differential radiative transfer equation. In order to overcome this difficulty, various typical numerical methods or solutions of radiative transfer equation (RTE) have been developed, including the discrete ordinates method (DOM) [3][4][5], the finite volume method (FVM) [5][6][7][8], the finite element method (FEM) [9][10][11], the spectral element method (SEM) [12,13], etc.…”

A deficiency problem of the least squares finite element method for solving radiative transfer in strongly inhomogeneous media

“…-6-The LSFEM was traditionally believed to be stable and accurate for solving partial differential equations in other disciplines [32][33][34], and hence for solving radiative transfer problems [20][21][22][23]. That is because the LSFEM owns two distinct advantages, such as (1) it is based on the principle of functional minimization, and (2) the stiff matrix produced by the LSFEM is symmetric and positive definite, which is a very good numerical property.…”

“…That is because the LSFEM owns two distinct advantages, such as (1) it is based on the principle of functional minimization, and (2) the stiff matrix produced by the LSFEM is symmetric and positive definite, which is a very good numerical property. Furthermore, it shows both accuracy and stability in many numerical tests [20][21][22][23]. It is noted that most of the tests are of radiative transfer in homogeneous media.…”

“…The schemes of FEM already studied for solving radiative transfer problems including the standard Galerkin scheme [18,19] and some advanced schemes, such as the least squares scheme [20][21][22][23], the streamline upwinding scheme [24,25], the scheme based on the even-parity form of the RTE (EPRTE) [26] and the scheme based on the second order radiative transfer equation (SORTE) [27]. The FEM based on the Galerkin scheme (GFEM) can be considered as the most basic version of FEM, which has been widely used in other disciplines [28][29][30].…”

“…Up till now, many numerical methods that are capable of solving radiative transfer in inhomogeneous absorbing, emitting and scattering media have been developed, such as the statistical method, e.g., the Monte Carlo method [9][10][11], and the deterministic methods based on discretization of the RTE, e.g., the discrete ordinate methods (DOM) [12][13][14], the finite volume method (FVM) [15][16][17] and the finite element method (FEM) [18][19][20][21][22][23][24][25][26][27], etc. The Monte Carlo method is by far the most versatile but it is very time consuming and is not a good choice for computational intensity applications, such as in optical topography and in solving other inverse radiative transfer problems.…”