A sweep-based low-rank method for the discrete ordinate transport equation

TL;DR

This paper introduces a sweep-based low-rank method for solving the discrete ordinate transport equation, significantly reducing memory and computational costs while maintaining accuracy in 2-D and 3-D geometries.

Contribution

It extends the dynamical low-rank approximation to the time-dependent radiation transport equation with a novel basis update and Galerkin integrator, improving stability and efficiency.

Findings

Reduces memory usage compared to full-rank methods

Decreases computational time in 2-D and 3-D problems

Maintains accuracy of the solution

Abstract

The dynamical low-rank (DLR) approximation is an efficient technique to approximate the solution to matrix differential equations. Recently, the DLR method was applied to radiation transport calculations to reduce memory requirements and computational costs. This work extends the low-rank scheme for the time-dependent radiation transport equation in 2-D and 3-D Cartesian geometries with discrete ordinates discretization in angle (SN method). The reduced system that evolves on a low-rank manifold is constructed via an unconventional basis update and Galerkin integrator to avoid a substep that is backward in time, which could be unstable for dissipative problems. The resulting system preserves the information on angular direction by applying separate low-rank decompositions in each octant where angular intensity has the same sign as the direction cosines. Then, transport sweeps and source iteration can efficiently solve this low-rank-SN system. The numerical results in 2-D and 3-D Cartesian geometries demonstrate that the low-rank solution requires less memory and computational time than solving the full rank equations using transport sweeps without losing accuracy.