标题： 积分形式中具有H矩阵的辐射传递的反射条件 显示英文标题

标题： Reflective Conditions for Radiative Transfer in Integral Form with H-Matrices

摘要： 在最近的一篇文章中，作者表明具有多个频率和散射的辐射传输方程可以表述为一个非线性积分系统。 在本文中，该公式被扩展以处理反射边界条件。 用于求解该系统的固定点方法被证明是单调的。 离散化是使用$P^1$有限元方法进行的。 卷积积分在网格的每个顶点处预先计算并存储在压缩的分层矩阵中，使用部分选主元的自适应交叉近似法。 然后，固定点迭代仅涉及矩阵与向量的乘积。 当所有内容都光滑时，该方法相对于顶点数量是$O(N\sqrt[3]{N}\ln N)$。 提出了一种数值实现并在两个示例上进行了测试。 由于与光线追踪有一些相似之处，因此编程较为复杂。 显示英文摘要

摘要： 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.