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.

翻译：动态低秩逼近（DLRA）是求解高维问题时降低计算成本、节省内存的新兴工具。本文提出并分析了一种半隐式动态低秩间断伽辽金（DLR-DG）方法，用于求解带有松弛算子的空间均匀动力学方程，该方程描述了背景介质对粒子的发射与吸收过程。DLRA与DG格式均可表示为伽辽金方程。为确保二者的一致性，引入加权DLRA，使得所得DLR-DG解在经典DG解空间的子空间中成为全离散DG格式的解。与经典DG方法类似，我们证明了所提出的DLR-DG方法的适定性。同时，我们识别了使DLR-DG解收敛到平衡态的条件。数值结果验证了理论发现。