Low-rank Methods for Radiation Transport Calculations

The numerical solution of the radiation transport equation (RTE) is challenging due to the high computational costs and the large memory requirements caused by the high-dimensional phase space. Here we detail an attempt to reduce the memory required, and computational cost of solving RTE, by applying the dynamical low-rank (DLR) method, where a memory savings of about an order of magnitude without sacrificing accuracy is observed. The DLR approximation is an efficient technique to approximate the solution to time-dependent matrix differential equations. The desired approximation has three components similar to factors in singular value decomposition, and each of them is solved by integrating the matrix differential equation projected onto the tangent space of the low-rank manifold. This talk presents our recent work that builds on the established DLR method and aims to enable low-rank schemes for practical radiation transport applications. We propose a high-order/low-order algorithm to overcome the conservation issues in the low-rank scheme by solving a low-order equation with closure terms computed via a high-order solution calculated using DLR. With the properly chosen rank, the high-order solution well approximates the closure term, and the low-order solution can be used to correct the conservation bias in the DLR evolution. This improvement goes a long way to making the method robust enough for a variety of physics applications. We also introduce a low-rank scheme with discrete ordinates discretization in angle (SN method). This low-rank-SN system allows for an efficient algorithm called “transport sweep,” which is highly desirable in computation. The derived low-rank SN equations can be cast into a triangular form in the same way as standard iteration techniques.