A stable multiplicative dynamical low-rank discretization for the linear Boltzmann-BGK equation

The method of dynamical low-rank approximation (DLRA) has recently been applied to various kinetic equations, showing a significant reduction in computational effort. In this paper, we apply this concept to the linear Boltzmann-Bhatnagar-Gross-Krook (Boltzmann-BGK) equation, which, due to its high dimensionality, is challenging to solve. Inspired by the special structure of the nonlinear 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 and 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.