Analysis of Stiffness in the Immersed Boundary Method and Implications for Time-Stepping Schemes

✍ Scribed by John M. Stockie; Brian R. Wetton

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 180 KB
• Volume: 154
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.1999.6297

✦ Synopsis

The immersed boundary method is known to exhibit a high degree of numerical stiffness associated with the interaction of immersed elastic fibres with the surrounding fluid. We perform a linear analysis of the underlying equations of motion for immersed fibres, and identify a discrete set of fibre modes which are associated solely with the presence of the fibre. This work generalises our results in a previous paper (1995, SIAM J. Appl. Math. 55, 1577) by incorporating the effect of spreading the singular fibre force over a finite "smoothing radius," corresponding to the approximate delta function used in the immersed boundary method. We investigate the stability of the fibre modes, their stiffness, and their dependence on the problem parameters, focusing on the influence of smoothing. We then extend the analytical results to include the effect of time discretisation, and draw conclusions about the time step restrictions on various explicit schemes, as well as the convergence rates for an iterative, semi-implicit method. We draw comparisons with computations and show how the results can be applied to help in choosing alternate time-stepping schemes that are specially tailored to handle the stiffness in immersed fibres. In particular, we present numerical results that show how fully explicit Runge-Kutta schemes perform in comparison with the best of the semi-implicit schemes currently in use.