2024/11/24 by Alec Dektor, Dektor, Alec, Lukas Einkemmer +1 · 9 citations
Computer Science · Engineering · Mathematics · #Computational Physics (physics.comp-ph) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Image and Signal Denoising Methods #Numerical Analysis (math.NA) #Sparse and Compressive Sensing Techniques #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.2411.15990
openalex publication_date 2024/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce two novel interpolatory dynamical low-rank (DLR) approximation methods for the efficient time integration of the Boltzmann-BGK equation. Both methods overcome limitations of classic DLR schemes based on orthogonal projections for nonlinear equations. In particular, we demonstrate that the proposed methods can efficiently compute solutions to the full Boltzmann-BGK equation without restricting to e.g. weakly compressible or isothermal flow. The first method we propose directly applies the recently developed interpolatory projector-splitting scheme on low-rank matrix manifolds. The second method is a variant of the rank-adaptive basis update and Galerkin scheme, where the Galerkin step is replaced by a collocation step, resulting in a new scheme we call basis update and collocate (BUC). Numerical experiments in both fluid and kinetic regimes demonstrate the performance of the proposed methods. In particular we demonstrate that the methods can be used to efficiently compute low-rank solutions in the six-dimensional (three spatial and three velocity dimensions) setting on a standard laptop.