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Interpolatory dynamical low-rank approximation for the 3+3d Boltzmann-BGK equation

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arxiv 2411.15990 v2 pith:X5CW3M6S submitted 2024-11-24 math.NA cs.NAphysics.comp-phphysics.flu-dyn

Alec Dektor , Lukas Einkemmer This is my paper · ORCID
classification math.NAcs.NAphysics.comp-phphysics.flu-dyn
keywords methodslow-rankboltzmann-bgkdemonstrateequationinterpolatoryschemeapproximation
verification ladder T0 review T1 audit T2 compute T3 formal
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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.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Asymptotic-Preserving Dynamical Low-Rank Method for the Stiff Nonlinear Boltzmann Equation

    math.NA 2025-02 conditional novelty 7.0 of 10

    A dynamical low-rank integrator (XL/sXL) for the stiff Boltzmann equation evaluates the collision operator r^2 times per step and is asymptotic-preserving in the fluid limit.

  2. An Asymptotic-Preserving Dynamical Low-Rank Semi-Lagrangian Method for Multiscale Linear Kinetic Transport Equations

    math.NA 2026-07 conditional novelty 6.0 of 10

    An asymptotic-preserving dynamical low-rank semi-Lagrangian solver with QDEIM angular sampling cuts the cost of multiscale kinetic transport simulations while preserving the diffusion limit.

  3. A Nodal Discontinuous Galerkin Method with Rank-Adaptive Velocity Space Representation for the Multiscale BGK Model

    math.NA 2025-08 conditional novelty 6.0 of 10

    A full-rank-in-space, low-rank-in-velocity nodal DG solver for 1d2v BGK is shown to be high-order, conservative, and asymptotic-preserving, with complexity linear in the velocity grid size.

  4. An Adaptive-rank Approach with Greedy Sampling for Multi-scale BGK Equations

    math.NA 2025-05 conditional novelty 6.0 of 10

    An adaptive-rank, greedy-sampling semi-Lagrangian solver with a macroscopic conservation correction is developed for the BGK equation and shown to be accurate, conservative, and conditionally asymptotic-preserving.

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