Analysis of Stiiness in the Immersed Boundary Method and Implications for Time-stepping Schemes

The immersed boundary method is known to exhibit a high degree of numerical sti ness, which is associated with the interaction of immersed elastic bres with the surrounding uid. We perform a linear analysis of the underlying equations of motion for immersed bres, and identify a discrete set of bre modes which are associated solely with the presence of the bre. These results are a generalisation of those in a previous paper (SIAM J. Appl. Math., 55(6):1577-1591, 1995) by including the e ect of spreading the singular bre force over a nite \smoothing radius," which corresponds to the approximate delta function used in the immersed boundary method. We investigate the stability of the bre modes, their sti ness and dependence on the problem parameters, and the e ect that smoothing has on the solution. The analytical results are then extended to include the e ects of time discretisation, and conclusions are drawn about the time step restrictions on various explicit time-stepping schemes, as well as the convergence rates for an iterative semi-implicit method. Comparisons are drawn with computations, and we show how the results can be applied to help in choosing alternate time-stepping schemes that are specially-tailored to handle the sti ness in immersed bres. In particular, we present numerical results that show how fully explicit Runge-Kutta schemes perform comparably with the best of semi-implicit schemes currently in use. 2