In this article, we study the physical mechanism of the Penning discharge, develop a full three-dimensional particle simulation software of high-quality algorithm (PIC), design and add the corresponding physical scenario of Monte Carlo collisions (MCC) module, and track electron, hydrogen molecular ion (H2+), hydrogen positive ion (H+), and tri-n-hydrogen ion (H3+) at the same time, and successfully develop a full three-dimensional electromagnetic PIC/MCC numerical algorithm. Combined with the Penning discharge model extensively studied in china, the algorithm is verified through simulation. The simulation results show that the use of effective filtering algorithm can suppress the electromagnetic numerical noise. Electron energy is of Maxwell distribution. Due to the radial drift and accelerate of electrons, the H2+ yield is larger at the top of the ion source.