  Dynamical low-rank approximation (DLRA) is an emerging tool for reducing
computational costs and provides memory savings when solving high-dimensional
problems. In this work, we propose and analyze a semi-implicit dynamical
low-rank discontinuous Galerkin (DLR-DG) method for the space homogeneous
kinetic equation with a relaxation operator, modeling the emission and
absorption of particles by a background medium. Both DLRA and the DG scheme can
be formulated as Galerkin equations. To ensure their consistency, a weighted
DLRA is introduced so that the resulting DLR-DG solution is a solution to the
fully discrete DG scheme in a subspace of the classical DG solution space.
Similar to the classical DG method, we show that the proposed DLR-DG method is
well-posed. We also identify conditions such that the DLR-DG solution converges
to the equilibrium. Numerical results are presented to demonstrate the
theoretical findings.
