Large-stepsize integrators for charged-particle dynamics over multiple time scales

TL;DR

This paper analyzes the performance of large-stepsize integrators, including Boris and variational methods, for simulating charged-particle dynamics in strong magnetic fields over multiple time scales, highlighting their strengths and limitations.

Contribution

It introduces and compares filtered variational integrators with traditional methods for large-step simulations of charged particles in magnetic fields, revealing their effectiveness and filtering requirements.

Findings

Filtered variational integrator performs well without filtering.

Boris algorithm requires filtering of initial velocities.

All methods conserve energy over long times.

Abstract

The Boris algorithm, a closely related variational integrator and a newly proposed filtered variational integrator are studied when they are used to numerically integrate the equations of motion of a charged particle in a non-uniform strong magnetic field, taking step sizes that are much larger than the period of the Larmor rotations. For the Boris algorithm and the standard (unfiltered) variational integrator, satisfactory behaviour is only obtained when the component of the initial velocity orthogonal to the magnetic field is filtered out. The particle motion shows varying behaviour over multiple time scales: fast Larmor rotation, guiding centre motion, slow perpendicular drift, near-conservation of the magnetic moment over very long times and conservation of energy for all times. Using modulated Fourier expansions of the exact and numerical solutions, it is analysed to which extent this behaviour is reproduced by the three numerical integrators used with large step sizes.