The magnetosphere of a rapidly rotating, strongly magnetized neutron star with aligned magnetic and rotational axes (parallel rotator) is modelled numerically. Including the radiation of the particles accelerated to relativistic energies as an efficient damping mechanism, we obtain a quasi-stationary self-consistent solution to this classical problem. The numerical simulation,which was started from the well-known vacuum solution, yields a global magnetospheric structure that can be characterized by two regions of oppositely charged particles, which eventually produce a relativistic pulsar wind, separated by a vacuum gap of considerable extent.