An Explicit Energy-Conserving Particle Method for the Vlasov-Fokker-Planck Equation

TL;DR

The paper addresses the challenge of numerically solving the Vlasov--Fokker--Planck equation with a nonlinear collision operator while preserving energy at the fully discrete level. It introduces an explicit particle method that combines a conservative discretization of the Fokker--Planck term with a second-order explicit time integrator featuring an accuracy-justified per-particle correction, and extends this framework to electromagnetic settings. The authors develop an optimization-based procedure to determine local temperature $T_p$ and bulk velocity $\mathbf{u}_p$ that enforce momentum and energy conservation, together with kernel-based regularization and spline interpolation for field and density reconstructions. Extensive benchmarks on electrostatic and electromagnetic problems, including linear and nonlinear Landau damping and the two-stream instability, demonstrate accurate energy conservation, robustness (especially in version 2), and competitive performance aided by cell-list optimization and GPU acceleration.

Abstract

We propose an explicit particle method for the Vlasov-Fokker-Planck equation that conserves energy at the fully discrete level. The method features two key components: a deterministic and conservative particle discretization for the nonlinear Fokker-Planck operator (also known as the Lenard-Bernstein or Dougherty operator), and a second-order explicit time integrator that ensures energy conservation through an accuracy-justifiable correction. We validate the method on several plasma benchmarks, including collisional Landau damping and two-stream instability, demonstrating its effectiveness.

Figures (12)

• Figure 1: Spatially homogeneous test. Order of convergence: $L^2$ error relative to the finest mesh $N_v=128$.
• Figure 2: Spatially homogeneous test. Velocity distributions at $t = 1$ and $10$ computed using the forward Euler, version 1, and version 2 schemes. Blue: histogram of the particle velocities. Orange: reconstructed distribution function.
• Figure 3: Spatially homogeneous test. Fractional change in total energy over time.
• Figure 4: Linear Landau damping test. Electric field energy trace up to time $t_{\text{final}} = 15$.
• Figure 5: Linear Landau damping test. Top: Fractional change in total energy. Bottom: Number of problematic particles. The means of ver1 and ver2 are computed as the total number of problematic particles divided by the number of time steps. The ratio is defined as the mean of ver1 to the mean of ver2.

Theorems & Definitions (4)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 3.1