In this paper, we propose and analyze a spherical harmonic discontinuous Galerkin (SH-DG) method for solving the radiative transfer equations with vacuum boundary conditions. To incorporate vacuum boundary conditions in spherical harmonic approximations, we first embed the original domain into a larger computational area of rectangular type with an extra pure absorbing layer and then establish a perturbation problem with a periodic condition at the boundary of the extended domain. Since the outflow radiative intensity at the outer boundary of the extended area can be made arbitrarily small by sufficiently increasing the magnitude of absorption in or the thickness of the absorbing layer, such a replacement of the boundary condition only causes a minimal difference between the solution of the perturbation problem and the original problem in the original domain, but will benefit the construction of the discretization scheme. Then based on the analysis of the perturbation problem and the SH-DG method for solving the radiative transfer equation with periodic boundary conditions, the well-posedness and the error estimates are derived for the approximation solution arising from the SH-DG method for solving the radiative transfer equation with vacuum boundary conditions. Numerical examples with both periodic and vacuum boundary conditions are included to validate the theoretical results.

在本文中，我们提出并分析了一种球谐谐波不连续伽勒金（SH-DG）方法，用于求解具有真空边界条件的辐射传递方程。为了将真空边界条件纳入球谐近似中，我们首先将原始域嵌入到具有额外纯吸收层的矩形较大的计算区域中，然后在扩展域的边界处建立具有周期条件的摄动问题。由于通过充分增大吸收层的吸收量或吸收层的厚度，可以任意地减小扩展区域的外边界处的流出辐射强度，因此，边界条件的这种替换仅引起扰动问题的解与原始域中的原始问题之间的最小差异，但是将有利于离散化方案的构造。然后在对摄动问题进行分析的基础上，采用SH-DG方法求解周期边界条件下的辐射传递方程，推导了SH-DG方法产生的近似解的适定性和误差估计。真空边界条件的辐射传递方程。包括具有周期性和真空边界条件的数值示例，以验证理论结果。然后在对摄动问题进行分析的基础上，采用SH-DG方法求解周期边界条件下的辐射传递方程，推导了SH-DG方法产生的近似解的适定性和误差估计。真空边界条件的辐射传递方程。包含周期性和真空边界条件的数值示例也包括在内，以验证理论结果。然后在对摄动问题进行分析的基础上，采用SH-DG方法求解周期边界条件下的辐射传递方程，推导了SH-DG方法产生的近似解的适定性和误差估计。真空边界条件的辐射传递方程。包含周期性和真空边界条件的数值示例也包括在内，以验证理论结果。