[Paper Review] Numerical Simulations to the Vlasov-Poisson System with a Strong Magnetic Field

Francis Filbet, Chang Yang|arXiv (Cornell University)|May 28, 2018

Gas Dynamics and Kinetic Theory22 references3 citations

TL;DR

This paper presents a high-order semi-implicit Particle-In-Cell method for the 3D Vlasov-Poisson system under strong magnetic fields, using time discretization that treats stiff magnetic terms implicitly and other terms explicitly. The scheme remains formally second-order consistent with the drift-kinetic asymptotic model even for large magnetic fields and coarse time steps, preserving energy and adiabatic invariants with relative errors below 10⁻³ over long simulations.

ABSTRACT

In this paper, we present an efficient Particle-In-Cell algorithm for the simulation of the three dimensional Vlasov-Poisson system in the presence of a strong external magnetic field. When the intensity of the magnetic field is sufficiently large and for any time step, the numerical scheme provides formally a consistent approximation of the drift-kinetic model, which corresponds to the asymptotic model. Numerical results show that this new Particle-In-Cell method is efficient and accurate large time steps.

Motivation & Objective

• To develop a stable and accurate numerical method for simulating the 3D Vlasov-Poisson system under strong external magnetic fields, where standard explicit schemes become inefficient due to stiffness.
• To extend asymptotically stable semi-implicit time discretization techniques from 2D to 3D, enabling long-time simulations with large time steps.
• To formally prove that the numerical scheme remains second-order accurate with respect to time step Δt and consistent with the drift-kinetic model when the magnetic field intensity ε⁻¹ is large.
• To validate the method numerically on both single-particle dynamics and the full Vlasov-Poisson system, demonstrating conservation of physical invariants.
• To enable efficient simulation of magnetized plasmas in fusion-relevant configurations, such as D-shaped tokamak domains, with realistic magnetic field profiles.

Proposed method

• A semi-implicit time discretization is applied to the characteristic curve system of the Vlasov-Poisson equation, where the stiff Lorentz force terms (proportional to ε⁻¹) are treated implicitly, while non-stiff terms are treated explicitly.
• High-order Runge-Kutta schemes (first, second, and third order) are constructed using a semi-implicit formulation to maintain accuracy and stability for large magnetic fields.
• The method is reformulated by dropping second-order terms in ε, leading to a discrete system that corresponds to the drift-kinetic model, ensuring asymptotic consistency.
• The Poisson equation is solved using standard spectral or finite difference methods on a Cartesian mesh, with charge density computed from particle weights via the Particle-In-Cell approach.
• The algorithm is implemented in a cylindrical geometry with a uniform magnetic field and later extended to non-uniform, radially increasing magnetic fields in a D-shaped domain.
• Numerical validation includes energy and adiabatic invariant conservation checks, as well as visualization of charge density evolution and filament formation.

Experimental results

Research questions

• RQ1 Can a semi-implicit time discretization maintain second-order accuracy in time for the 3D Vlasov-Poisson system under strong magnetic fields?
• RQ2 Does the proposed method remain stable and accurate even when the time step is much larger than the cyclotron period (i.e., Δt ≫ ε)?
• RQ3 Is the numerical solution formally consistent with the drift-kinetic asymptotic model for ε ≪ 1, even when the magnetic field is large?
• RQ4 Can the method preserve key physical invariants—such as total energy and adiabatic invariant—over long-time simulations with coarse time grids?
• RQ5 How does the method perform in realistic 3D configurations, such as a D-shaped tokamak domain with non-uniform magnetic fields?

Key findings

• The semi-implicit scheme maintains second-order accuracy in time even for large magnetic fields, with formal consistency to the drift-kinetic model verified analytically.
• For ε = 0.01 and Δt = 0.5, the relative variation in total energy remained below 10⁻³ over long simulations, demonstrating excellent conservation properties.
• The adiabatic invariant μ was conserved with relative error of order 10⁻³ over long times, confirming the method's ability to capture slow-scale dynamics accurately.
• Snapshots of the charge density evolution showed the formation of persistent filamentary structures and halo-like propagation, consistent with turbulent transport in magnetized plasmas.
• The method successfully simulated the merging of two plasma density peaks in a D-shaped domain under a non-uniform magnetic field, showing robust confinement and physical realism.
• The algorithm efficiently filters fast oscillations (cyclotron motion) when Δt ≫ ε, allowing large time steps without loss of accuracy on slow-scale variables.