Abstract: Tsunamis pose a hazard that may strike a coastal population within a short amount of time. To effectively forecast and warn for tsunamis, extremely fast simulations are needed. However, until recently such urgent tsunami simulations have been infeasible in the context of early warning and even for high-resolution rapid post-event assessment. The implementation of efficient tsunami numerical codes using Graphical Processing Units (GPUs) has now allowed much faster simulations, which have opened a new avenue for carrying out simulations Faster Than Real Time (FTRT). This paper discusses the need for urgent computing in computational tsunami science, and presents workflows for two applications, namely FTRT itself and Probabilistic Tsunami Forecasting (PTF). PTF relies on a very high number of FTRT simulations addressing forecasting uncertainty, whose full quantification will be made more and more at reach with the advent of exascale computing resources.