Boundary conditions for reaction-diffusion systems in an Immersed Boundary framework

The Immersed Boundary Method (IBM) is widely used in the simulation of systems involving fluid-structure interaction. The ability to simulate problems involving complex geometry using a simple, Eulerian grid has led to the use of the IBM for the study of many biomechanical and biophysical systems. However, the study of chemical systems often necessitates the ability to impose Neumann or Robin boundary conditions at irregular locations on the Eulerian grid. Here, we present a method for imposing Neumann and Robin boundary conditions within an IBM framework. The method requires solving a larger, augmented linear system. However, this sytem is well-conditioned and can be solved with a small number of iterations of a standard Kylov method without preconditioning. We also present applications and limitations of our methodology.