Gyrotrons for fusion applications are microwave vacuum tubes that are capable to produce an output power in the megawatt range at long pulses up to continuous wave (CW) and at frequencies above 100 GHz. That is possible due to the working principle of gyrotrons which allows using cavities with a very large electrical size (in the order of several cm) compared to the operating wavelength (in the order of a few mm). This mandatory requirement for high output power is a challenge in simulating the interaction between the electromagnetic (EM) field and the electron beam in a gyrotron resonator. Due to this, the simulation of the electron interaction in gyrotrons are typically carried out by using computer codes which make use of the very specific properties of the EM problem to simplify the calculations. At KIT, a new code names “SimpleRick” is under development. A fast-time scale Particle-in-Cell (PIC) method is implemented to complement the classical models used for gyrotron simulation. The PIC code introduces significantly fewer assumptions than the classical model and may therefore represent more physical details. For example, in contrast to the classical models, the new model can represent non-symmetric electron beams. In this work, the numerical implementation and the performance of this PIC model are verified and a new method for the calculation of the eigenvalues of coaxial gyrotron resonators is shown in more detail.