The low-rank approximation is a complexity reduction technique to approximate a tensor or a matrix with a reduced rank, which has been applied to the simulation of high dimensional problems to reduce the memory required and computational cost. In this work, a dynamical low-rank approximation method is developed for the time-dependent radiation transport equation in 1-D and 2-D Cartesian geometries. Using a finite volume discretization in space and a spherical harmonics basis in angle, we construct a system that evolves on a low-rank manifold via an operator splitting approach. Numerical results on five test problems demonstrate that the low-rank solution requires less memory and computational time than solving the full rank equations with the same accuracy. It is furthermore shown that the low-rank algorithm can obtain high-fidelity results by increasing the number of basis functions while keeping the rank fixed.

低秩逼近是一种复杂性降低技术，用于逼近具有降低秩的张量或矩阵，该技术已应用于高维问题的仿真，以减少所需的内存和计算成本。在这项工作中，针对一维和二维笛卡尔几何中与时间有关的辐射传输方程，开发了一种动态低秩逼近方法。使用空间中的有限体积离散和角度中的球谐函数，我们构建了一个通过算子拆分方法在低秩流形上演化的系统。关于五个测试问题的数值结果表明，与以相同的精度求解满秩方程相比，低秩求解所需的内存和计算时间更少。