A stable multiplicative dynamical low-rank discretization for the linear Boltzmann-BGK equation
This work provides a more efficient numerical method for solving the linear Boltzmann-BGK equation, which is beneficial for researchers and engineers working with high-dimensional kinetic equations.
This paper addresses the computational challenge of the high-dimensional linear Boltzmann-BGK equation by applying a rank-adaptive dynamical low-rank approximation (DLRA) scheme. The method, which uses a multiplicative splitting of the distribution function, basis update & Galerkin integrator, and basis augmentation, is shown to be accurate and efficient compared to solving the full system.
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.