This paper presents a positive and asymptotic preserving scheme for the nonlinear gray radiative transfer equations. The scheme is constructed by combining the filtered spherical harmonics ($F{P}_N$) method for the discretization of angular variable and with the framework of the unified gas kinetic scheme (UGKS) for the spatial- and time-discretization. The constructed scheme is almost free of ray effects and can also mitigate oscillations in the spherical harmonics ($P_N$) approximation. Moreover, it can be shown that the current scheme is asymptotic preserving. Consequently, in the optically thick regimes the current scheme can exactly capture the solution of the diffusion limit equation without requiring the cell size being smaller than the photon's mean free path, while the solution in optically thin regimes can also be well resolved in a natural way. In addition, the $F{P}_N$ angular discretization induces a natural macro-micro decomposition, with this help we can obtain the sufficient conditions that guarantee the positivity of the radiative energy density and material temperature. Then, a linear scaling limiter is given to enforce that sufficient conditions. With the process of such construction, we finally obtain a scheme, called the $PPF{P}_N$-based UGKS scheme, that is positive and asymptotic preserving. Various numerical experiments are included to validate the robustness, positive- and asymptotic-preserving property as well as the property of almost ray effect free.

本文提出了非线性灰色辐射传递方程的正渐近保持方案。该方案是通过将滤波后的球谐函数（$F{磷}_N$) 用于角度变量离散化的方法，以及用于空间和时间离散化的统一气体动力学方案 (UGKS) 的框架。构建的方案几乎没有射线效应，并且还可以减轻球谐函数中的振荡（${磷}_N$) 近似。此外，可以证明当前方案是渐近保持的。因此，在光学厚区域中，当前方案可以准确地捕获扩散极限方程的解，而无需单元尺寸小于光子的平均自由程，而在光学薄区域中的解也可以以自然的方式很好地解决. 除此之外$F{磷}_N$角度离散化引发了自然的宏观-微观分解，通过这种帮助，我们可以获得保证辐射能量密度和材料温度为正的充分条件。然后，给出一个线性缩放限制器来强制执行该充分条件。通过这样的构建过程，我们最终得到了一个方案，称为$磷磷F{磷}_N$基于 UGKS 方案，即正渐近保持。包括各种数值实验以验证稳健性、正和渐近保持特性以及几乎无射线效应的特性。