Two Dimensional Immersed Interface Method for Moving Interfaces: An Implementation with Applications to Biological Interface Problems

Immersed interface problems usually arise when modeling the dynamics between two different materials, like water and oil, or the same material at different states, such as water and ice. The modeling and computational simulation of these problems have multiple applications ranging from ice-melting to biological interface problems, like cell, bubble and biofilm deformation. The immersed interface method is a powerful second order accurate numerical method to model such problems. Since implementing this method is not trivial, it is hard for scientists in the biological areas or others to take advantage of it. Part of the ongoing goal of this project is to make this code as general as possible, open access, and user friendly, so that interested scientists without the technical knowledge or the time to implement it can employ it in their own research. In the present work, we have implemented the immersed interface method for 2-D heat equations with fixed or moving immersed interfaces and a singular forcing term along the interface. Simulations are presented and compared with known analytical solutions to verify the algorithm. Sample problems, like ice-melting, are fully implemented, and possible biological applications are discussed. Further extensions of this work are intended to be used in modeling the interaction of shock-waves and biological interfaces. Potential future biomedical applications are: shock-wave therapy for biofilm breaking and localized drug delivery using high intensity focused ultrasound to break drug filled micro-bubbles.