A comparison of implicit solvers for the immersed boundary equations

Explicit time discretizations of the immersed boundary method are known to require small timesteps to maintain stability. A number of implicit methods have been introduced to alleviate this restriction to allow for a more efficient method, but many of these methods still have a stability restriction on the timestep. Furthermore, almost no comparisons have appeared in the literature of the relative computational costs of the implicit methods and the explicit method. A recent paper [E.P. Newren, A.L. Fogelson, R.D. Guy, R.M. Kirby, Unconditionally stable discretizations of the immersed boundary equations, J. Comput. Phys. 222 (2007) 702–719.] addressed the confusion over stability of immersed boundary discretizations. This paper identified the cause of instability in previous immersed boundary discretizations as lack of conservation of energy and introduced a new semi-implicit discretization proven to be unconditionally stable, i.e., it has bounded discrete energy. The current paper addresses the issue of the efficiency of the implicit solvers. Existing and new methods to solve implicit immersed boundary equations are described. Systematic comparisons of computational cost are presented for a number of these solution methods for our stable semi-implicit immersed boundary discretization and an explicit discretization for two distinct test problems. These comparisons show that two of the implicit methods are at least competitive with the explicit method on one test problem and outperform it on the other test problem in which the elastic stiffness of the boundary does not dictate the timescale of the fluid motion.