Using integral equations and the immersed interface method to solve immersed boundary problems with stiff forces

We propose a fast, explicit numerical method for computing approximations for the immersed boundary problem in which the boundaries that separate the fluid into two regions are stiff. In the numerical computations of such problems, one frequently has to contend with numerical instability, as the stiff immersed boundaries exert large forces on the local fluid. When the boundary forces are treated explicitly, prohibitively small time-steps may be required to maintain numerical stability. On the other hand, when the boundary forces are treated implicitly, the restriction on the time-step size is reduced, but the solution of a large system of coupled non-linear equations may be required. In this work, we develop an efficient method that combines an integral equation approach with the immersed interface method. The present method treats the boundary forces explicitly. To reduce computational costs, the method uses an operator-splitting approach: large time-steps are used to update the non-stiff advection terms, and smaller substeps are used to advance the stiff boundary. At each substep, an integral equation is computed to yield fluid velocity local to the boundary; those velocity values are then used to update the boundary configuration. Fluid variables are computed over the entire domain, using the immersed interface method, only at the end of the large advection time-steps. Numerical results suggest that the present method compares favorably with an implementation of the immersed interface method that employs an explicit time-stepping and no fractional stepping.