A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele–Shaw cell

In this paper, we present a time and space rescaling scheme for the computation of moving interface problems. The idea is to map time–space such that the interfaces can evolve exponentially fast in the new time scale while the area/volume enclosed by the interface remains unchanged. The rescaling scheme significantly reduces the computation time (especially for slow growth), and enables one to accurately simulate the very long-time dynamics of moving interfaces. We then implement this scheme in a Hele–Shaw problem, examine the dynamics for a number of different injection fluxes, and present the largest and most pronounced viscous fingering simulations to date.