Boris and Exponential Integrators in the Theory of Particles Interacting with Magnetic Turbulence

The interaction of electrically charged particles with magnetic fields is a fundamental problem in several areas of physics. An example is the motion of energetic particles through a magnetized plasma. The most accurate and reliable way to explore theoretically the interactions between particles and fields is via test-particle simulations. In such simulations one creates the turbulent magnetic field and solves the Newton-Lorentz equation numerically by employing an integration scheme. In the current article we discuss exponential integrators and derive systematically from this the Rodrigues scheme as well as the famous Boris integrator. For an approach where one creates the magnetic field anew at each time step, both integrators are overall comparable. In theory the Rodrigues approach should be more accurate due to the fact that the occurring matrix exponential is evaluated without further approximations. Practically, both methods provide very similar results. It is argued in the current article that a Rodrigues based integrator is a very strong alternative because for the specific problem discussed here, it does not require longer computing times.