  We present a novel numerical scheme for simulating the motion of relativistic
charged particles in magnetospheres of compact objects, typically filled with
highly magnetized collisionless plasmas. The new algorithm is based on a
dynamic switch between the full system of equations of motion and a guiding
center approximation. The switch between the two formulations is based on the
magnetization of the plasma particles, such that the dynamics are accurately
captured by the guiding center motion even when the gyro-frequency is
under-resolved by the time step. For particles with a large gyro-radius, due to
acceleration in, e.g., reconnecting current sheets, the algorithm adaptively
switches to solve the full equations of motion instead. The new scheme is
directly compatible with standard Particle-in-Cell codes, and is readily
applicable in curved spacetimes via a dedicated covariant formulation. We test
the performance of the coupled algorithm by evolving charged particles in
electromagnetic configurations of reconnecting current sheets in magnetized
plasma, obtained from special- and general-relativistic Particle-in-Cell
simulations. The new coupled pusher is capable of producing highly accurate
particle trajectories even when the time step is many orders of magnitude
larger than the gyro-period, substantially reducing the restrictions of the
temporal resolution.
