In this thesis, we develop efficient adaptive rescaling schemes to investigate interface instabilities associated with moving interface... Show moreIn this thesis, we develop efficient adaptive rescaling schemes to investigate interface instabilities associated with moving interface problems. The idea of rescaling is to map the current time-space onto a new time-space frame such that the interfaces evolve at a chosen speed in the new frame. We couple the rescaling idea with boundary integral method to demonstrate the efficiency of the rescaling idea, though it can be applied to Cartesian-grid based method in general. As an example, we use the Hele-Shaw problem to examine the efficiency of the rescaling scheme. First, we apply the rescaling scheme to a slowly expanding interface. In the new frame, the evolution is dramatically accelerated, while the underlying physics remains unchanged. In particular, at long times numerical results reveal that there exist nonlinear, stable, self-similarly evolving morphologies. The rescaling idea can also be used to simulate the fast shrinking interface, e.g. the Hele-Shaw problem with a time dependent gap. In this case, the rescaling scheme slows down the interface evolution in the new frame to remove the severe time step constraint that makes the long-time simulations prohibitive. Finally, we study an analytical solution to the stability of the interface of the Hele-Shaw problem, assuming a small surface tension under a time dependent flux Q(t). Following [116, 109], we find the motions of daughter singularity ζd and simple singularity ζ0 do not depend on the flux Q(t). We also find a criterion to identify the relation between ζ0 and ζd. Ph.D. in Applied Mathematics, July 2017 Show less