Second-order radiative transfer equation and its properties of numerical solution using the finite-element method

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.