  Abridged. Kinetic plasma simulations are nowadays commonly used to study a
wealth of non-linear behaviours and properties in laboratory and space plasmas.
In particular, in high-energy physics and astrophysics, the plasma usually
evolves in ultra-strong electromagnetic fields produced by intense laser beams
for the former or by rotating compact objects such as neutron stars and black
holes for the latter. In these ultra-strong electromagnetic fields, the
gyro-period is several orders of magnitude smaller than the timescale on which
we desire to investigate the plasma evolution. Some approximations are required
like for instance artificially decreasing the electromagnetic field strength
which is certainly not satisfactory. The main flaw of this downscaling is that
it cannot reproduce particle acceleration to ultra-relativistic speeds with
Lorentz factor above $\gamma \approx 10^3-10^4$. In this paper, we design a new
algorithm able to catch particle motion and acceleration to Lorentz factor up
to $10^{15}$ or even higher by using Lorentz boosts to special frames where the
electric and magnetic field are parallel. Assuming that these fields are
locally uniform in space and constant in time, we solve analytically the
equation of motion in a tiny region smaller than the length scale of the
spatial and temporal gradient of the field.
