  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.
