We investigate controls on tsunami generation and propagation in the near-field of great megathrust earthquakes using a series of numerical simulations of subduction and tsunamigenesis on the Sumatran forearc. The Sunda megathrust here is advanced in its seismic cycle and may be ready for another great earthquake. We calculate the seafloor displacements and tsunami wave heights for about 100 complex earthquake ruptures whose synthesis was informed by reference to geodetic and stress accumulation studies. Remarkably, results show that, for any near-field location: (1) the timing of tsunami inundation is independent of slip-distribution on the earthquake or even of its magnitude, and (2) the maximum wave height is directly proportional to the vertical coseismic displacement experienced at that location. Both observations are explained by the dominance of long wavelength crustal flexure in near-field tsunamigenesis. The results show, for the first time, that a single estimate of vertical coseismic displacement might provide a reliable short-term forecast of the maximum height of tsunami waves.