  A modification of the standard Boris algorithm, called filtered Boris
algorithm, is proposed for the numerical integration of the equations of motion
of charged particles in a strong non-uniform magnetic field in the asymptotic
scaling known as maximal ordering. With an appropriate choice of filters,
second-order error bounds in the position and in the parallel velocity, and
first-order error bounds in the normal velocity are obtained with respect to
the scaling parameter. The proof compares the modulated Fourier expansions of
the exact and the numerical solutions. Numerical experiments illustrate the
error behaviour of the filtered Boris algorithm.
