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

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

Dynamical low-rank approximation for Marshak waves

Marshak waves are temperature waves which can arise from the background radiation in a material. A core limitation in the simulation of these temperature waves is the high-dimensional phase space of the radiation solution, which depends on time, the spatial position as well as the direction of flight. To obtain computationally efficient methods, we propose to use dynamical low-rank approximation (DLRA) which is a model order reduction method that dynamically determines and adapts dominant modes of the numerical solution. This is done by projecting the original dynamics onto the tangent space of the low-rank manifold. In this work, we investigate discontinuous Galerkin discretizations for two robust time integrators. By performing the derivation of the DLRA evolution equations on the continuous level, we are able to apply the needed slope limiter on the low-rank factors instead of the full solution. The efficiency of the method is presented through computational results for a Marshak wave originating from a heated wall

Geometric Numerical Integration (hybrid meeting)

The topics of the workshop included interactions between geometric numerical integration and numerical partial differential equations; geometric aspects of stochastic differential equations; interaction with optimisation and machine learning; new applications of geometric integration in physics; problems of discrete geometry, integrability, and algebraic aspects