  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.
