  Two phase flows that include phase transition, especially phase creation,
with a sharp interface remain a challenging task for numerics. We consider the
isothermal Euler equations with phase transition between a liquid and a vapor
phase. The phase interface is modeled as a sharp interface and the mass
transfer across the phase boundary is modeled by a kinetic relation. Existence
and uniqueness results were proven in Ref. \cite{Hantke2019a}. Using sharp
interfaces for simulating nucleation and cavitation results in the grid
containing tiny cells that are several orders of magnitude smaller than the
remaining grid cells. This forces explicit time stepping schemes to take tiny
time steps on these cells. As a remedy we suggest an explicit implicit domain
splitting where the majority of the grid cells is treated explicitly and only
the neighborhood of the tiny cells is treated implicitly. We use dual time
stepping to solve the resulting small implicit systems. Our numerical results
indicate that the new scheme is robust and provides significant speed-up
compared to a fully explicit treatment.
