Low-Rank Methods for Radiation Transport Calculations

<p>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<i>_N</i>) 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. </p><p>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. </p><p>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<i>_N</i>) 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.</p>