The numerical method of dynamical low-rank approximation (DLRA) has recently been applied to various kinetic equations showing a significant reduction of the computational effort. In this paper, we apply this concept to the linear Boltzmann-Bhatnagar-Gross-Krook (Boltzmann-BGK) equation which due its high dimensionality is challenging to solve. Inspired by the special structure of the non-linear Boltzmann-BGK problem, we consider a multiplicative splitting of the distribution function. We propose a rank-adaptive DLRA scheme making use of the basis update & Galerkin integrator and combine it with an additional basis augmentation to ensure numerical stability, for which an analytical proof is given and a classical hyperbolic Courant-Friedrichs-Lewy (CFL) condition is derived. This allows for a further acceleration of computational times and a better understanding of the underlying problem in finding a suitable discretization of the system. Numerical results of a series of different test examples confirm the accuracy and efficiency of the proposed method compared to the numerical solution of the full system.

翻译：动力学低秩近似（DLRA）数值方法近年来已被应用于各类动力学方程，显著降低了计算成本。本文将该方法应用于高维求解极具挑战性的线性 Boltzmann-Bhatnagar-Gross-Krook（Boltzmann-BGK）方程。受非线性 Boltzmann-BGK 问题特殊结构的启发，我们考虑对分布函数进行乘性分裂。我们提出了一种采用基更新与 Galerkin 积分器的秩自适应 DLRA 格式，并结合额外的基扩充技术以保证数值稳定性——对此我们给出了解析证明，并推导出经典的 Courant-Friedrichs-Lewy（CFL）双曲型条件。这使得计算时间得以进一步加速，并有助于更深入理解在寻找系统合适离散化方案时的本质问题。一系列不同数值算例的结果表明，与完整系统的数值解相比，所提方法在精度和效率上均具有优越性。