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.
Bibliographical noteFunding Information:
The authors thank Prof. M. Shelley and P. Fast for stimulating discussions and for sending a version of the HLS94 code  with improved data structures. The authors thank the National Science Foundation (Division of Mathematical Science) and the University of Minnesota Office of Sponsored Projects for partial support. S. Li thanks Yubao Zhen for technical discussions. The authors also acknowledge the generous computing resources from the Network and Academic Computing Services (NACS) at University of California at Irvine (UCI), and the hospitality of Biomedical Engineering Department at UCI.
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- Boundary integral method
- Moving boundary problems
- Saffman-Taylor instability