A low-rank method for two-dimensional time-dependent radiation transport calculations

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 than solving the full rank equations with the same accuracy. It is furthermore shown that the low-rank algorithm can obtain high-fidelity results at a moderate extra cost by increasing the number of basis functions while keeping the rank fixed.