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

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