In order to probe rare events in life- and material sciences as well as high-energy physics, the intensity of the beam provided by particle accelerators is raised to levels where the self-interaction of the beam particles due to electromagnetic repulsion, the so-called space-charge effect, becomes a dominant factor. To complement the existing simulation methods, we developed a fully analytic and self-consistent space-charge model that is valid for transverse Gaussian beams and include non-linear forces and cross-plane coupling. Based on this model, we developed a simulation code that provides a quick means of parameter studies. In this presentation, I will present the concept of the analytical model and show simulation results of the development of a beam in a test ring. Further, I show examinations of passive spectator particles, revolving around the previously simulated beam core and investigate their dynamic behavior as well as distortions in phase space due to non-linear forces.