A modified Crank-Nicolson scheme for the Vlasov-Poisson system with a strong external magnetic field

TL;DR

This work develops an augmented Crank-Nicolson time discretization within a PIC framework to solve the Vlasov-Poisson system under a strong, inhomogeneous external magnetic field. By introducing an additional energy variable and an auxiliary velocity, the scheme decouples fast gyromotion from slow guiding-center dynamics, achieving uniform accuracy as the small parameter ε→0. The authors show the method is asymptotically consistent with the guiding-center model, preserves (or nearly preserves) key invariants, and remains stable for large Δt in long-time simulations, including diocotron instabilities and vortex interactions. Numerical results demonstrate superior performance over Brackbill–Forslund–Vu and Ricketson–Chacón CN schemes, and the approach is poised for extension to 3D and more complex field configurations.

Abstract

We propose and study a Particle-In-Cell (PIC) method based on the Crank-Nicolson time discretization for the Vlasov-Poisson system with a strong and inhomogeneous external magnetic field with fixed direction, where we focus on the motion of particles in the plane orthogonal to the magnetic field. In this regime, the time step can be subject to stability constraints related to the smallness of Larmor radius and plasma frequency [21]. To avoid this limitation, our approach is based on numerical schemes [9, 10, 12], providing a consistent PIC discretization of the guiding-center system taking into account variations of the magnetic field. We carry out some theoretical proofs and perform several numerical experiments to validate the method and its underlying concepts.

Figures (14)

• Figure 4.1: One single particle motion: Numerical errors of discrete solution $\mathbf{x}_{\varepsilon , \Delta t}$, approximated by several schemes: \ref{['scheme:Brackbill']}, \ref{['scheme:Chacon']}, IMEX2L and \ref{['scheme:modified_CN']}, with reference solution $\mathbf{x}_\varepsilon$ of \ref{['eq:ODE_system']} for various $\varepsilon > 0$ and $\Delta t > 0$.
• Figure 4.2: One single particle motion: Numerical errors of discrete solution $\mathbf{x}_{\varepsilon , \Delta t}$, approximated by several schemes: \ref{['scheme:Brackbill']}, \ref{['scheme:Chacon']}, IMEX2L and \ref{['scheme:modified_CN']}, with guiding center solution $\mathbf{y}$ of \ref{['eq:guiding_center_system']} for various $\varepsilon > 0$ and $\Delta t > 0$.
• Figure 4.3: One single particle motion: Numerical errors of discrete solution $e_{\varepsilon , \Delta t}$, approximated by several schemes: \ref{['scheme:Brackbill']}, \ref{['scheme:Chacon']}, IMEX2L and \ref{['scheme:modified_CN']}, with guiding center solution $g$ of \ref{['eq:guiding_center_system']} for various $\varepsilon > 0$ and $\Delta t > 0$.
• Figure 4.4: One single particle motion: Trajectory of particle approximated by several schemes: \ref{['scheme:Brackbill']}, \ref{['scheme:Chacon']}, IMEX2L and \ref{['scheme:modified_CN']} with $\varepsilon = 0.01$, $\Delta t = 0.1$ and final time $T = 30s$.
• Figure 4.5: One single particle motion: Variation of energy approximated by several schemes: \ref{['scheme:Brackbill']}, \ref{['scheme:Chacon']}, IMEX2L and \ref{['scheme:modified_CN']} with $\varepsilon = 0.01$, $\Delta t = 0.1$ and final time $T = 30s$.

Theorems & Definitions (10)

• Proposition 2.1: Asymptotic behavior $\varepsilon \rightarrow 0$ with a fixed $\Delta t$
• proof
• Proposition 2.2: Asymptotic behavior $\varepsilon \rightarrow 0$ with a fixed $\Delta t$
• proof
• Proposition 2.3: Asymptotic behavior $\varepsilon \rightarrow 0$ with a fixed $\Delta t$
• proof
• Remark 2.4
• Proposition 3.1: Consistency in the limit $\varepsilon \rightarrow 0$ for a fixed $\Delta t$
• proof
• Remark 4.1