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We present a novel numerical method for simulating the generation and propagation of surface gravity waves by vertical seabed displacements. The cornerstone of our method is the computation of a time dependent conformal map which incorporates the time dependent geometry of the seabed and the wave profile along the free surface. This enables us to handle general geometric configurations of the seabed and the wave. As benchmark we reproduce the results of Hammack on tsunami generation and propagation. Our results show that Hammack's linear theory accurately predicts wave generation. However, as the velocity of the sea bed displacement increases, nonlinear effects become increasingly noticeable. Notably, when the seabed uplift occurs rapidly, the following nonlinear dynamics of the wave differ significantly from the linear dynamics usually associated with tsunami propagation.