Due to its integro-differential nature, deriving schemes for numerically solving the radiative transfer equation (RTE) is challenging. Most solvers are efficient in specific scenarios: structured grids, simulations with low-scattering materials... In this paper, a full solver, from the discretization of the steady-state monochromatic RTE to the solution of the resulting system, is derived.

Using a mixed matrix-ready/matrix-free approach, our solver is able to discretize and solve a 45.7 billion unknown problem on 27 thousand processes in three minutes for a full physics involving scattering, absorption, and reflection. Because of the high dimensionality of the continuous equation, the linear system would have had more than $6\times {10}^{15}$ nonzero entries if assembled explicitly. Our approach allows for large memory gains by only storing lower dimension reference matrices.

The finite element-based solver is wrapped around open-source software, FreeFEM for discretization, PETSc for linear algebra, and hypre for the algebraic multigrid infrastructure. Overall, deterministic results are presented on arbitrarily-decomposed unstructured grids for radiative transfer problems with scattering, absorbing, and reflecting heterogeneities on up to 27 thousand processes.

由于其积分微分性质，用于数值求解辐射传递方程（RTE）的推导方案具有挑战性。大多数求解器在特定情况下都是有效的：结构化网格，使用低散射材料的仿真...在本文中，从稳态单色RTE的离散化到所得系统的求解，得出了一个完整的求解器。

使用混合矩阵就绪/无矩阵方法，我们的求解器能够在3分钟内离散化和求解277万个过程中的457亿个未知问题，从而实现涉及散射，吸收和反射的完整物理过程。由于连续方程的高维性，线性系统将具有$6\times {10}^{15}$非零条目（如果显式组装）。我们的方法通过仅存储较低维的参考矩阵来实现较大的内存增益。

基于有限元的求解器包括开源软件，用于离散化的FreeFEM，用于线性代数的PETSc和用于代数多网格基础结构的hypre。总体而言，在任意分解的非结构化网格上给出了确定性结果，该网格用于多达27,000个过程中具有散射，吸收和反射异质性的辐射传递问题。