While general purpose integrators like the Runge-Kutta methods can be applied to a wide range of problems, specialized algorithms like the Velocity-Verlet and Boris methods are tailored for specific types of systems, such as those encountered in molecular dynamics and charged particle dynamics, respectively. While Velocity-Verlet offers very good energy conservation properties for systems with position-dependent forces, it is not applicable for problems involving charged particles in magnetic fields, as the Lorentz force depends on velocity. The Boris algorithm, on the other hand, is specifically designed to handle the velocity-dependent nature of the magnetic force and is widely used in plasma physics and charged particle simulations. It delivers a similar numerical accuracy as the Runge-Kutta methods, while being phase-space preserving and requiring a fraction of the computational cost.

All algorithms including Forward Euler integration improve their accuracy with decreasing time step size, as the local truncation error per step is proportional to a power of the time step size $h$. However, due to limitations in computational resources, purpose built algorithms like Boris algorithm are needed to efficiently simulate complex systems over long time periods with sufficient accuracy. Thus, Boris algorithm is the preferred choice for simulating Penning traps.

If there is no knowledge about the system to be simulated, we have shown near perfect results for the RK4 method at a slightly increased computational cost. Furthermore, RK4 was energy conserving in our simulations. If phase-space conservation is not of uttermost importance, using Runge-Kutta methods is a good choice for general purpose simulations.