The main challenge of precise point positioning (PPP) applications is the long convergence time of typically a half hour, or even more, to achieve centimeter accuracy. Even when the multi-constellation is involved and ambiguity resolution is implemented, it still requires about ten minutes. It is becoming a hot spot to incorporate the low Earth orbit (LEO) satellite constellation for enhancing the Global Navigation Satellite System (GNSS), named here as LEO-enhanced GNSS (LeGNSS). In this system, the LEO satellites cannot only receive GNSS signals, but also serve as GNSS satellites by transmitting similar navigation signals to the ground users, but with higher signal strength and much faster geometric change due to their low altitude. As a result, the convergence time of PPP is expected to be shortened to a few minutes, or even seconds. Simulation software is developed to simulate GNSS and LEO observations for ground stations taking into account tropospheric delay, satellite clock errors, observation noises, as well as other error sources. Then the number of visible satellites, the geometry dilution of precision (GDOP), and the convergence time of the kinematic mode of PPP are evaluated on a global scale compared to those of GNSS systems. The simulation results show that LeGNSS can decrease the PPP convergence to 5 min. If there are more LEO satellites included in the LeGNSS, it is expected that the initialization of PPP can be further shortened.