The Hele-Shaw problem is a classical example for studying interface dynamics or systems driven out of equilibrium. In this talk, I will first discuss numerical issues related to long-time computation of moving interfaces. Then I will present a time adaptive scheme for efficiently computing the dynamics of a moving interface. The idea is to design a time-space mapping such that in the new time scale, the interfaces can evolve at arbitrarily fast or slow speed. We then show numerical examples and present largest (expanding interface) and smallest (shrinking interface) Hele-Shaw simulation up to date.