Penning traps are useful appliances to capture charged particles without contact to any matter surrounding the particles \cite[11]{vogel_particle_2024}. This property of Penning traps is especially useful in the study of antimatter, as any contact to matter would lead to the direct annihilation of the antimatter. Therefore, Penning traps have become an essential tool in the antimatter research. Experiments like ALPHA-g \cite{bertsche_prospects_2018} and AEgIS \cite{scampoli_aegis_2014} at CERN have successfully deployed Penning traps for such applications.

To plan future antimatter experiments it is essential to be able to simulate Penning traps. Without the ability of the behavior of particles within such a trap, a precise planning of the Penning trap and antimatter experiment is not possible. Key criterion for a successful simulation in this context is accuracy of the numerical simulation in terms of precise trajectories and key statistical properties of the captured particles, e.g. the kinetic energy. A second metric that is very important to the usefulness of such numerical simulations is the computational performance of the simulation as this decides whether the simulation is even possible or not with finite computing resources. To find the optimal balance between the finite computing power available and an accurate result, different integration algorithms will be compared to solve the ordinary differential equations at play in the Penning Trap. 

In \cref{sec:methods} the methods employed in the simulation will be expanded on. In detail the implementation of the source code will be discussed as well as the specific Penning trap used for the study, introduced. Thirdly the algorithms behind the numerical integration methods will be explained. \Cref{sec:results_and_discussion} will show the results obtained from our implementation and discuss their practical implications. This section will be split into four parts expanding on the accuracy of the trajectories, the energy conservation in the system, the computational performance of the different algorithms as well as a discussion on the limit of many particles in the trap. Lastly in \cref{sec:conclusion} our findings will be summarized and consequences of the results derived.