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Published November 10, 2008 | public
Journal Article Open

Removing the Stiffness of Elastic Force from the Immersed Boundary Method for the 2D Stokes Equations


The immersed boundary method has evolved into one of the most useful computational methods in studying fluid structure interaction. On the other hand, the immersed boundary method is also known to suffer from a severe timestep stability restriction when using an explicit time discretization. In this paper, we propose several efficient semi-implicit schemes to remove this stiffness from the immersed boundary method for the two-dimensional Stokes flow. First, we obtain a novel unconditionally stable semi-implicit discretization for the immersed boundary problem. Using this unconditionally stable discretization as a building block, we derive several efficient semi-implicit schemes for the immersed boundary problem by applying the small scale decomposition to this unconditionally stable discretization. Our stability analysis and extensive numerical experiments show that our semi-implicit schemes offer much better stability property than the explicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit schemes can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme, but with a much better stability property.

Additional Information

© 2008 Elsevier Inc. Received 16 June 2007; received in revised form 21 December 2007; accepted 2 March 2008. Available online 10 March 2008. Special Issue Celebrating Tony Leonard's 70th Birthday. We would like to thank Profs. Charles Peskin and Hector Ceniceros for a number of stimulating discussions on the Immersed Boundary method. The research was in part supported by DOE under the DOE Grant DE-FG02-06ER25727 and by NSF under the NSF FRG Grant DMS-0353838, ITR Grants ACI-0204932 and DMS-0713670.

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August 22, 2023
August 22, 2023