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Relativistic strongly magnetized plasmas are produced in laboratories thanksto state-of-the-art laser technology but can naturally be found around compactobjects such as neutron stars and black holes. Detailed studies of thebehaviour of relativistic plasmas require accurate computations able to catchthe full spatial and temporal dynamics of the system. Numerical simulations ofultra-relativistic plasmas face severe restrictions due to limitations in themaximum possible Lorentz factors that current algorithm can reproduce to goodaccuracy. In order to circumvent this flaw and repel the limit to$\gamma\approx10^9$, we design a new fully implicit scheme to solve therelativistic particle equation of motion in an external electromagnetic fieldusing a three dimensional Cartesian geometry. We show some examples ofnumerical integrations in constant electromagnetic fields to prove theefficiency of our algorithm. The code is also able to follow the electric driftmotion for high Lorentz factors. In the most general case of spatially andtemporally varying electromagnetic fields, the code performs extremely well asshown by comparison with exact analytical solutions for the relativisticelectrostatic Kepler problem as well as for linearly and circularly polarizedplane waves.

An fully implicit scheme for numerical integration of the relativistic particle equation of motion