Low-Rank Methods for Radiation Transport Calculations

This dissertation seeks to reduce the computational cost in radiation transport calculations using dynamical low-rank approximation (DLR) methods, a complexity reduction technique to approximate a tensor or a matrix with a reduced rank. A DLR approximation method is developed for the time-dependent radiation transport equation in 1-D and 2-D Cartesian geometries. A low-rank system that evolves on a low-rank manifold via an operator-splitting approach is constructed using a finite volume (FV) method in space and a spherical harmonics (P_N) basis in angle. Numerical results demonstrate that the low-rank solution requires less memory than solving the full-rank equations with the same accuracy. Furthermore, the low-rank algorithm can obtain much better results at a moderate extra cost by refining the discretization while keeping the rank fixed.

The DLR method does not preserve the number of particles, which limits its practicability. This conservation issue is addressed by solving a low-order equation with closure terms coupled to a low-rank approximation with the high-order solution. The high-order solution approximates the closure term well, and the low-order solution corrects the low-rank evolution's conservation bias. The so-called high-order / low-order (HOLO) algorithm is demonstrated that overcomes the conservation difficulty while the computational efficiency and accuracy are preserved.

The low-rank scheme is extended for the time-dependent radiation transport equation in 2-D and 3-D Cartesian geometries with discrete ordinates (S_N) discretization in angle. The reduced system that evolves on a low-rank manifold is constructed via an “unconventional” basis update & 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.