Second-Order Radiative Transfer Equation and Its Properties of Numerical Solution Using the Finite-Element Method

Abstract

The original radiative transfer equation is a first-order integrodifferential equation, which can be taken as a convection-dominated equation. The presence of the convection term may cause nonphysical oscillation of solutions. This type of instability can occur in many numerical methods, including the finite-difference method and the finite-element method, if no special stability treatment is used. To overcome this problem, a second-order radiative transfer equation is derived, which is a diffusion-type equation similar to the heat conduction equation for an anisotropic medium. The consistency of the second-order radiative transfer equation with the original radiative transfer equation is demonstrated. The perturbation characteristics of error are analyzed and compared for both the first-and second-order equations. Good numerical properties are found for the second-order radiative transfer equation. To show the properties of the numerical solution, the standard Galerkin finite-element method is employed to solve the second-order radiative transfer equation. Four test problems are taken as examples to check the numerical properties of the second-order radiative transfer equation. The results show that the standard Galerkin finite-element solution of the second-order radiative transfer equation is numerically stable, efficient, and accurate.

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

“…By discretization with central difference like numerical method, such as the FEM and the meshless methods, the second order forms of radiative transfer equation shows better numerical properties than the RTE. Besides the even parity formulation of the RTE, there are three second order forms of the radiative transfer equations have been proposed, the SORTE [27,36], the LSORTE [35] and the MSORTE [35]. The performance of the second order radiative transfer equations have been studied by the author in several papers [35].…”

“…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]. Former studies [23,27] reported that the GFEM is instable in solving some radiative transfer problems and will produce results with spurious oscillations. An interpretation to this instability is based on an analogy with the solution of the convection diffusion equation.…”

“…The streamline upwinding scheme introduces an undetermined tuning parameter that is not easy to be chosen. The FEM based on the EPRTE and the SORTE [27] have singularity problems in dealing with inhomogeneous media where some locations have very small/zero extinction coefficient due to the exist of the reciprocal of extinction coefficient in the formulation. The LSFEM has not all these problems.…”

“…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.…”