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A structure-preserving collisional particle method for the Landau kinetic equation

Kai Du, Lei Li, Yongle Xie, Yang Yu

TL;DR

Addressing the numerical solution of the Landau kinetic equation, the paper develops a structure-preserving stochastic particle method that models grazing collisions as diffusion between paired particles. It introduces an exact temporal discretization based on spherical Brownian motion, ensuring the discrete-time particle distribution matches the continuous model and achieving $O(N)$ complexity per step. The method preserves mass, momentum, energy, and entropy dissipation, exhibits strong long-time stability, and extends to the non-homogeneous Vlasov–Poisson–Landau equation via a collision-transport splitting with an energy-conserving PIC step. This yields an efficient and accurate framework for simulating collisional plasmas with potential applicability to broader kinetic models.

Abstract

In this paper, we propose and implement a structure-preserving stochastic particle method for the Landau equation. The method is based on a particle system for the Landau equation, where pairwise grazing collisions are modeled as diffusion processes. By exploiting the unique structure of the particle system and a spherical Brownian motion sampling, the method avoids additional temporal discretization of the particle system, ensuring that the discrete-time particle distributions exactly match their continuous-time counterparts. The method achieves $O(N)$ complexity per time step and preserves fundamental physical properties, including the conservation of mass, momentum and energy, as well as entropy dissipation. It demonstrates strong long-time accuracy and stability in numerical experiments. Furthermore, we also apply the method to the spatially non-homogeneous equations through a case study of the Vlasov--Poisson--Landau equation.

A structure-preserving collisional particle method for the Landau kinetic equation

TL;DR

Addressing the numerical solution of the Landau kinetic equation, the paper develops a structure-preserving stochastic particle method that models grazing collisions as diffusion between paired particles. It introduces an exact temporal discretization based on spherical Brownian motion, ensuring the discrete-time particle distribution matches the continuous model and achieving complexity per step. The method preserves mass, momentum, energy, and entropy dissipation, exhibits strong long-time stability, and extends to the non-homogeneous Vlasov–Poisson–Landau equation via a collision-transport splitting with an energy-conserving PIC step. This yields an efficient and accurate framework for simulating collisional plasmas with potential applicability to broader kinetic models.

Abstract

In this paper, we propose and implement a structure-preserving stochastic particle method for the Landau equation. The method is based on a particle system for the Landau equation, where pairwise grazing collisions are modeled as diffusion processes. By exploiting the unique structure of the particle system and a spherical Brownian motion sampling, the method avoids additional temporal discretization of the particle system, ensuring that the discrete-time particle distributions exactly match their continuous-time counterparts. The method achieves complexity per time step and preserves fundamental physical properties, including the conservation of mass, momentum and energy, as well as entropy dissipation. It demonstrates strong long-time accuracy and stability in numerical experiments. Furthermore, we also apply the method to the spatially non-homogeneous equations through a case study of the Vlasov--Poisson--Landau equation.
Paper Structure (10 sections, 3 theorems, 47 equations, 6 figures, 6 algorithms)

This paper contains 10 sections, 3 theorems, 47 equations, 6 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. The collisional particle system and its properties
  3. Discretization schemes
  4. Extension to Vlasov--Poisson--Landau equation
  5. Numerical experiments
  6. 2D BKW solution for Maxwell molecules
  7. 2D anisotropic solution with a singular kernel
  8. 3D BKW solution for Maxwell molecules
  9. A Vlasov--Poisson--Landau example with Coulomb potential
  10. Conclusion

Key Result

Proposition 2.1

Under the above setting, the quantities remain constant in time.

Figures (6)

  • Figure 1: Time evolution of relative $L_2$ error, energy and entropy for different $N$, where (a)(b) shows the results of both SBM and EM schemes and (c) show the results of SBM scheme.
  • Figure 2: Convergence rate (left) and CPU time (right) of SBM scheme
  • Figure 3: Time evolution of relative $L_2$ error, energy for different $N$ , where (a)(b) shows the results of both SBM and EM schemes.
  • Figure 4: Convergence order (left) and CPU time (right) of SBM scheme
  • Figure 5: Time evolution of relative $L_2$ error and energy for different $N$.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Proposition 2.1: Pathwise conservation of total momentum and energy
  • proof
  • Proposition 2.2: Entropy dissipation
  • Remark 2.1
  • Proposition 3.1
  • proof
  • Remark 3.1