Dynamic rupture propagations on seismic faults are numerically simulated based on finite element method, finite-difference method and boundary integral equation method. Boundary integral equation method provides the accurate stress field near the fault surface, while the method is just applied for a simple homogeneous medium. Domain based method, such as finite element method, is applicable to complex media, while low accuracies for the stress field near the fault surface are expected. We apply a combination method of integral equation method and finite element method. Fault rupture process and excited wave propagations are numerically calculated in the simple half space 2-D P-SV medium.