A separable and asymptotic-preserving dynamical low-rank method for the Vlasov–Poisson–Fokker–Planck system

2026/01/31 by Shiheng Zhang, Jingwei Hu
Computer Science · Mathematics · Physics and Astronomy · #Discretization #Dynamical systems theory #Field (mathematics) #Model Reduction and Neural Networks #Operator (biology) #Property (philosophy) #Separable space #Stochastic Gradient Optimization Techniques #Tensor decomposition and applications #cs.NA #math.NA #msc:35Q83 #msc:35Q84 #msc:65F30 #msc:65M06 #msc:65M12

paper · pdf · doi:10.1051/m2an/2026047

openalex publication_date 2026/05/18 · openalex created_date 2026/05/19 · arxiv created 2026/05/27 · openalex updated_date 2026/08/04 · arxiv updated 2026/08/05

Abstract

We present a dynamical low-rank (DLR) method for the Vlasov–Poisson–Fokker–Planck (VPFP) system. Our main contributions are twofold: (i) a conservative spatial discretization of the Fokker–Planck operator that factors into velocity-only and space-only components, enabling efficient low-rank projection, and (ii) a time discretization within the DLR framework that properly handles stiff collisions. We propose both first-order and second-order low-rank IMEX schemes. For the first-order scheme, we prove an asymptotic-preserving (AP) property when the field fluctuation is small. Numerical experiments demonstrate accuracy, robustness, and AP property at modest ranks.

Citations

• Interpolatory Dynamical Low-Rank Approximation: Theoretical Foundations and Algorithms
• On the stability of the low-rank projector-splitting integrators for hyperbolic and parabolic equations
• Asymptotic-Preserving Dynamical Low-Rank Method for the Stiff Nonlinear Boltzmann Equation
• A review of low-rank methods for time-dependent kinetic simulations
• Interpolatory dynamical low-rank approximation for the 3+3d Boltzmann-BGK equation
• Asymptotic--preserving and energy stable dynamical low-rank approximation
• Efficient dynamical low-rank approximation for the Vlasov-Ampère-Fokker-Planck system
• On the stability of robust dynamical low-rank approximations for hyperbolic problems
• A rank-adaptive robust integrator for dynamical low-rank approximation
• An unconventional robust integrator for dynamical low-rank approximation
• Variational Asymptotic Preserving Scheme for the Vlasov-Poisson-Fokker-Planck System
• Hypocoercivity and Uniform Regularity for the Vlasov-Poisson-Fokker-Planck System with Uncertainty and Multiple Scales
• A projector-splitting integrator for dynamical low-rank approximation
• LOW AND HIGH FIELD SCALING LIMITS FOR THE VLASOV– AND WIGNER–POISSON–FOKKER–PLANCK SYSTEMS
• Runaway Phenomena and Fluid Approximation Under High Fields in Semiconductor Kinetic Theory
• A Review of Mathematical Topics in Collisional Kinetic Theory