A spatial second-order scheme for the nonlinear radiative transfer equations is introduced in this paper. The discretization scheme is based on the filtered spherical harmonics ($FP_N$) method for the angular variable and the unified gas kinetic scheme (UGKS) framework for the spatial and temporal variables respectively. In order to keep the scheme positive and second-order accuracy, firstly, we use the implicit Monte Carlo linearization method [6] in the construction of the UGKS numerical boundary fluxes. Then, by carefully analyzing the constructed second-order fluxes involved in the macro-micro decomposition, which is induced by the $FP_N$ angular discretization, we establish the sufficient conditions that guarantee the positivity of the radiative energy density and material temperature. Finally, we employ linear scaling limiters for the angular variable in the $P_N$ reconstruction and for the spatial variable in the piecewise linear slopes reconstruction respectively, which are shown to be realizable and reasonable to enforce the sufficient conditions holding. Thus, the desired scheme, called the $PPFP_N$-based UGKS, is obtained. Furthermore, in the regime $\epsilon\ll 1$ and the regime $\epsilon=O(1)$, a simplified spatial second-order scheme, called the $PPFP_N$-based SUGKS, is presented, which possesses all the properties of the non-simplified one. Inheriting the merit of UGKS, the proposed schemes are asymptotic preserving. By employing the $FP_N$ method for the angular variable, the proposed schemes are almost free of ray effects. To our best knowledge, this is the first time that spatial second-order, positive, asymptotic preserving and almost free of ray effects schemes are constructed for the nonlinear radiative transfer equations without operator splitting. Various numerical experiments are included to validate the properties of the proposed schemes.

翻译：本文提出了针对非线性辐射传输方程的空间二阶格式。该离散格式基于角变量滤波球谐函数（$FP_N$）方法，以及时空变量统一气体动理学格式（UGKS）框架。为保持格式的正保真性与二阶精度，首先在UGKS数值边界通量构建中采用隐式蒙特卡罗线性化方法[6]。其次，通过深入分析由$FP_N$角离散引发的宏微观分解中涉及的二阶通量构造，建立了保证辐射能量密度与材料温度正性的充分条件。最后，分别对$P_N$重构中的角变量与分段线性斜率重构中的空间变量采用线性缩放限制器，验证了这些限制器可实现且合理,从而确保充分条件成立。由此得到所需格式——基于$PPFP_N$的UGKS。进一步地，在$\epsilon\ll 1$和$\epsilon=O(1)$区域内,提出了简化空间二阶格式——基于$PPFP_N$的SUGKS，该格式保留了非简化格式的所有特性。继承UGKS优点，所提格式具备渐近保持性。通过采用$FP_N$方法处理角变量，所提格式几乎不存在射线效应。据我们所知，这是首次在不采用算子分裂的情况下，为非线性辐射传输方程构建兼具空间二阶精度、正保真性、渐近保持性及几乎无射线效应的格式。文中包含多项数值实验以验证所提格式的性能。