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UGKS and UGKWP Methods for Multiscale Simulation of Electrostatic Plasma in Quasineutral and Hydrodynamic Limits

Zhigang Pu, Kun Xu

TL;DR

This work tackles multiscale electrostatic plasma dynamics governed by both the Debye length $\lambda$ and the mean-free-path; it develops asymptotic-preserving (AP) schemes by coupling collision and transport fluxes within the UGKS and UGKWP frameworks and by adopting a Reformulated Poisson Equation (RPE). The authors introduce the VP-BGK model and its reformulated counterpart, deriving the quasineutral and hydrodynamic/quasineutral limits to ensure correct asymptotic behavior. The resulting UGKS–RPE and UGKWP–RPE methods provide unified-preserving and AP properties, respectively, through a coupled flux formulation and a mixed equilibrium-particle representation that naturally handles vanishing $\lambda$ and stiff collision terms $\tau$. Numerical tests on nonlinear and linear Landau damping and bump-on-tail instabilities demonstrate robust performance across collisionless to collisional regimes and across under-resolved Debye lengths, highlighting potential for efficient simulation of complex hypersonic plasma flows. The approach offers a scalable, accurate tool for multiscale electrostatic plasmas with practical impact in fusion, space, and semiconductor processing contexts, where $\lambda$ and $l_{mfp}$ pose significant computational challenges.

Abstract

This study extends the Unified Gas-Kinetic Scheme (UGKS) and the Unified Gas-Kinetic Wave-Particle (UGKWP) method for electrostatic plasma modeling, ensuring the correct asymptotic limits with respect to both the Debye length and the mean free path. By coupling collision and transport processes within the numerical flux, the proposed approach effectively removes the hydrodynamic-limit constraint associated with the mean free path. In addition, a reformulated Poisson equation, coupled with the macroscopic moment equations, is introduced to overcome the inefficiency of the standard Poisson formulation in the quasineutral regime. The accuracy and asymptotic consistency of the proposed schemes are verified through several benchmark tests, including linear and nonlinear Landau damping and the bump-on-tail instability. The results demonstrate that the methods robustly capture plasma dynamics across hydrodynamic and quasineutral regimes, without resolution constraints imposed by either the Debye length or the mean free path.

UGKS and UGKWP Methods for Multiscale Simulation of Electrostatic Plasma in Quasineutral and Hydrodynamic Limits

TL;DR

This work tackles multiscale electrostatic plasma dynamics governed by both the Debye length and the mean-free-path; it develops asymptotic-preserving (AP) schemes by coupling collision and transport fluxes within the UGKS and UGKWP frameworks and by adopting a Reformulated Poisson Equation (RPE). The authors introduce the VP-BGK model and its reformulated counterpart, deriving the quasineutral and hydrodynamic/quasineutral limits to ensure correct asymptotic behavior. The resulting UGKS–RPE and UGKWP–RPE methods provide unified-preserving and AP properties, respectively, through a coupled flux formulation and a mixed equilibrium-particle representation that naturally handles vanishing and stiff collision terms . Numerical tests on nonlinear and linear Landau damping and bump-on-tail instabilities demonstrate robust performance across collisionless to collisional regimes and across under-resolved Debye lengths, highlighting potential for efficient simulation of complex hypersonic plasma flows. The approach offers a scalable, accurate tool for multiscale electrostatic plasmas with practical impact in fusion, space, and semiconductor processing contexts, where and pose significant computational challenges.

Abstract

This study extends the Unified Gas-Kinetic Scheme (UGKS) and the Unified Gas-Kinetic Wave-Particle (UGKWP) method for electrostatic plasma modeling, ensuring the correct asymptotic limits with respect to both the Debye length and the mean free path. By coupling collision and transport processes within the numerical flux, the proposed approach effectively removes the hydrodynamic-limit constraint associated with the mean free path. In addition, a reformulated Poisson equation, coupled with the macroscopic moment equations, is introduced to overcome the inefficiency of the standard Poisson formulation in the quasineutral regime. The accuracy and asymptotic consistency of the proposed schemes are verified through several benchmark tests, including linear and nonlinear Landau damping and the bump-on-tail instability. The results demonstrate that the methods robustly capture plasma dynamics across hydrodynamic and quasineutral regimes, without resolution constraints imposed by either the Debye length or the mean free path.

Paper Structure

This paper contains 18 sections, 2 theorems, 65 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Kinetic Model Equations
  3. Vlasov-Poisson-BGK model
  4. Reformulated Vlasov-Poisson-BGK model
  5. Asymptotic limit
  6. Quasineutral limit
  7. Hydrodynamical and quasineutral limit
  8. UGKS and UGKWP method
  9. General Framework
  10. Electrical field evolution scheme
  11. UGKS method
  12. UGKWP method
  13. Analysis
  14. Numerical studies
  15. Nonlinear Landau damping
  16. ...and 3 more sections

Key Result

Lemma 1

The Vlasov-Poisson-BGK system $\mathbf{P}^{\lambda,\tau}$ is equivalent to the reformulated Vlasov-Poisson-BGK system $\mathbf{RP}^{\lambda,\tau}$ if and only if the initial potential $\phi_0$ and its time derivative $\phi^{'}_0$ satisfy the following initial conditions:

Figures (14)

  • Figure 1: Nonlinear Landau damping with different Knudsen numbers by UGKS-RPE and UGKWP-RPE: $\lambda = 1, \Delta x = 0.1, \text{CFL}=0.9$. The Figure shows the time evolution of electrostatic energy.
  • Figure 2: Nonlinear Landau damping with different Knudsen numbers by UGKS-RPE, from left to right are $Kn=\infty, 1, 10^{-4}$. $\lambda = 1, \Delta x = 0.1, \text{CFL}=0.9$. The Figure shows the phase diagram at different times.
  • Figure 3: Nonlinear Landau damping with different Knudsen numbers by UGKWP-RPE, from left to right are $Kn=\infty, 1, 10^{-4}$. $\lambda = 1, \Delta x = 0.1, \text{CFL}=0.9$. The Figure shows the phase diagram at different times.
  • Figure 4: Nonlinear Landau damping with different Knudsen numbers by UGKS-RPE: $\lambda = 1, \Delta x = 0.1, \text{CFL}=0.9$. The figure shows distribution at velocity space $u$ for different time at $x=L/2$, from top-left to bottom-right are t = 1, 4, 8, 15.
  • Figure 5: Bump on tail instability by UGKS-RPE and UGKWP-PRE, resolved case: $\lambda = 1, \Delta x = 0.08, \text{CFL} = 0.9$ at t=40. Left: time evolution of electrostatic energy. Right: time evolution of timestep.
  • ...and 9 more figures

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Lemma 2
  • proof