Unconditionally stable discretizations of the immersed boundary equations

The immersed boundary (IB) method is known to require small timesteps to maintain stability when solved with an explicit or approximately implicit method. Many implicit methods have been proposed to try to mitigate this timestep restriction, but none are known to be unconditionally stable, and the observed instability of even some of the fully implicit methods is not well understood. In this paper, we prove that particular backward Euler and Crank-Nicolson-like discretizations of the nonlinear immersed boundary terms of the IB equations in conjunction with unsteady Stokes Flow can yield unconditionally stable methods. We also show that the position at which the spreading and interpolation operators are evaluated is not relevant to stability so as long as both operators are evaluated at the same location in time and space. We further demonstrate through computational tests that approximate projection methods (which do not provide a discretely divergence-free velocity field) appear to have a stabilizing influence for these problems; and that the implicit methods of this paper, when used with the full Navier-Stokes equations, are no longer subject to such a strict timestep restriction and can be run up to the CFL constraint of the advection terms.