In a recent article the authors showed that the radiative Transfer equations with multiple frequencies and scattering can be formulated as a nonlinear integral system. In the present article, the formulation is extended to handle reflective boundary conditions. The fixed point method to solve the system is shown to be monotone. The discretization is done with a $P^1$ Finite Element Method. The convolution integrals are precomputed at every vertices of the mesh and stored in compressed hierarchical matrices, using Partially Pivoted Adaptive Cross-Approximation. Then the fixed point iterations involve only matrix vector products. The method is $O(N\sqrt[3]{N}\ln N)$, with respect to the number of vertices, when everything is smooth. A numerical implementation is proposed and tested on two examples. As there are some analogies with ray tracing the programming is complex.

翻译：在近期的一篇文章中，作者证明了具有多频率和散射效应的辐射传输方程可以表述为一个非线性积分系统。本文将该表述扩展至反射边界条件。用于求解该系统的不动点方法被证明具有单调性。离散化过程采用$P^1$有限元方法。卷积积分在每个网格顶点处预先计算，并通过部分主元自适应交叉逼近技术以压缩分层矩阵形式存储。此后，不动点迭代仅涉及矩阵向量乘积。当所有变量光滑时，该方法关于顶点数量的计算复杂度为$O(N\sqrt[3]{N}\ln N)$。本文提出了一种数值实现方案，并在两个算例上进行了测试。由于该方法与光线追踪存在若干相似性，编程过程较为复杂。