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A Separable and Asymptotic-Preserving Dynamical Low-Rank Method for the Vlasov--Poisson--Fokker--Planck System

Shiheng Zhang, Jingwei Hu

TL;DR

This work develops a dynamical low-rank framework for the Vlasov–Poisson–Fokker–Planck system, addressing both high dimensionality and stiffness. A conservative, separable discretization of the Fokker–Planck operator enables efficient low-rank projection without full-tensor reconstruction, and a projector-splitting approach evolves the low-rank manifold via K–S–L steps. The authors introduce first- and second-order low-rank IMEX time integrators, with a rigorous AP analysis for the first-order scheme in the small-field-fluctuation regime and comprehensive numerical validation. Numerical experiments in 1D1V confirm robust AP behavior and accurate macroscopic limits at modest ranks, demonstrating the method’s potential for scalable kinetic simulations.

Abstract

We present a dynamical low-rank (DLR) method for the Vlasov--Poisson--Fokker--Planck (VPFP) system. Our main contributions are two-fold: (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.

A Separable and Asymptotic-Preserving Dynamical Low-Rank Method for the Vlasov--Poisson--Fokker--Planck System

TL;DR

This work develops a dynamical low-rank framework for the Vlasov–Poisson–Fokker–Planck system, addressing both high dimensionality and stiffness. A conservative, separable discretization of the Fokker–Planck operator enables efficient low-rank projection without full-tensor reconstruction, and a projector-splitting approach evolves the low-rank manifold via K–S–L steps. The authors introduce first- and second-order low-rank IMEX time integrators, with a rigorous AP analysis for the first-order scheme in the small-field-fluctuation regime and comprehensive numerical validation. Numerical experiments in 1D1V confirm robust AP behavior and accurate macroscopic limits at modest ranks, demonstrating the method’s potential for scalable kinetic simulations.

Abstract

We present a dynamical low-rank (DLR) method for the Vlasov--Poisson--Fokker--Planck (VPFP) system. Our main contributions are two-fold: (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.
Paper Structure (27 sections, 3 theorems, 96 equations, 8 figures, 2 algorithms)

This paper contains 27 sections, 3 theorems, 96 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Dynamical low-rank formulation for the VPFP system
  3. K-step (update the spatial basis).
  4. S-step (update the coefficient matrix).
  5. L-step (update the velocity basis).
  6. Separable discretization of the Fokker--Planck operator and its low-rank projection
  7. Construction of the separable stencil
  8. Low-rank projection
  9. K-step.
  10. S-step.
  11. L-step.
  12. Time discretization and the fully discrete low-rank scheme
  13. A first-order scheme
  14. A first-order low-rank IMEX scheme
  15. A second-order scheme
  16. ...and 12 more sections

Key Result

Proposition 5.1

Consider the decomposition $f = \rho M + g$, where the fluctuation satisfies the local mass conservation property $\langle g, 1 \rangle_v^h = 0$. There exists a constant $\gamma_h > 0$ such that the Fokker--Planck operator satisfies the coercivity estimate: where the constant $\gamma_h$ is defined as:

Figures (8)

  • Figure 1: Time-step convergence and AP property for the first-order (top) and second-order (bottom) low-rank IMEX schemes.
  • Figure 2: Mixed-regime test: comparison of macroscopic density (top row) and electric field (bottom row) between the full-tensor reference solution (solid lines) and the first-order low-rank IMEX scheme (circles) at $t=0.10$ and $t=0.30$.
  • Figure 3: Phase plots in the mixed regime test: full tensor, low-rank, and absolute difference.
  • Figure 4: Spatial profiles for the mixed-regime test: (a) the Knudsen number $\varepsilon(x)$; (b) and (c) the pointwise AP error $\mathcal{E}_{AP}(x,t)$ at $t=0.10$ and $t=0.30$, respectively.
  • Figure 5: Bump-on-tail test: macroscopic density and electric field in the kinetic ($t=0.10$, $\varepsilon_0=1$) and fluid ($t=0.50$, $\varepsilon_0=10^{-6}$) regimes.
  • ...and 3 more figures

Theorems & Definitions (9)

  • Proposition 5.1: Discrete coercivity
  • proof
  • Proposition 5.2: Fluctuation--based projection bound
  • proof
  • Theorem 5.3: AP property up to spatial fluctuation
  • proof
  • Remark 5.4
  • Remark 5.5
  • Remark 5.6