Two-dimensional leapfrog scheme for trajectories of relativistic charged particles in static axisymmetric electric and magnetic field

A method for the calculation of two-dimensional particle trajectories is proposed in this work. It makes use of the cylindrical symmetry and the simplification of the static electric field, so that there should be no systematic error for the centered large-orbit rotations nor for the acceleration or deceleration in a uniform electric field. The method also shows a lower error level than the standard Boris method in many cases. Typical applications of this method are for example, electron microscopes, electron guns and collectors of gyro-devices as well as of other vacuum tubes, which can be described in axisymmetric cylindrical coordinates. Besides, the proposed method enforces the conservation of canonical angular momentum by construction, which is expected to show its advantages in the simulation of cusp electron guns and other components relying on non-adiabatic transitions in the externally applied static magnetic field.