Magnetic resonance elastography (MRE) can noninvasively quantify tissue elastic properties. There are two kinds of dynamic actuation in MRE, namely harmonic excitation and transient excitation. Due to the complexity of human organs and tissues, the propagation of elastic waves suffers from reflection, diffraction and other interfering effects. In this study, a numerical platform was established to simulate transient MRE. Furthermore, an algorithm of elasticity reconstruction was developed in accordance with time of arrival. A series of numerical experiments were carried out for harmonic and transient MREs. The results show that transient MRE has good anti-interference ability on reflection and diffraction, and the developed algorithm can perform elasticity reconstruction for transient MRE successfully.