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 () 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 (IMC) linearization method in the construction of the UGKS numerical boundary fluxes. This is an essential point in the construction. Then, by carefully analyzing the constructed second-order fluxes involved in the macro-micro decomposition, which is induced by the 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 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 -based UGKS, is obtained. Furthermore, we can show that in the regime and the regime , the second-order fluxes can be simplified. And, a simplified spatial second-order scheme, called the -based SUGKS, is thus 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 method for the angular variable, the proposed schemes are almost free of ray effects. Moreover, the above-mentioned way of imposing the positivity would not destroy both AP and second-order accuracy properties. 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. Therefore, this paper improves our previous work on the first-order scheme which could not be directly extended to high order, while keeping the solution positive. Various numerical experiments are included to validate the properties of the proposed schemes.

中文翻译：

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非线性辐射传递方程的空间二阶正渐近保滤PN格式

本文介绍了非线性辐射传输方程的空间二阶格式。离散化方案基于角度变量的滤波球谐函数 () 方法和分别针对空间和时间变量的统一气体动力学方案 (UGKS) 框架。为了保持格式的正二阶精度，首先，我们在UGKS数值边界通量的构造中使用隐式蒙特卡罗（IMC）线性化方法。这是施工中的一个要点。然后，通过仔细分析由角度离散引起的宏微观分解所构造的二阶通量，我们建立了保证辐射能量密度和材料温度为正值的充分条件。最后，我们分别对重建中的角度变量和分段线性斜率重建中的空间变量采用线性缩放限制器，这被证明是可实现的且合理的，以强制满足充分条件。因此，获得了所需的方案，称为基于 UGKS。此外，我们可以证明，在 和 区域中，二阶通量可以被简化。并且，由此提出了一种简化的空间二阶方案，称为基于 的 SUGKS，它具有非简化空间二阶方案的所有属性。继承了UGKS的优点，所提出的方案是渐近保持的。通过采用角度变量的方法，所提出的方案几乎没有射线效应。此外，上述施加正性的方式不会破坏 AP 和二阶精度属性。据我们所知，这是第一次在没有算子分裂的情况下为非线性辐射传递方程构造空间二阶、正、渐近保持且几乎无射线效应的方案。因此，本文改进了我们之前对不能直接推广到高阶的一阶格式的工作，同时保持解为正。包括各种数值实验来验证所提出方案的属性。