Efficient Semi-Implicit Schemes For Immersed Boundary Method

The Immersed Boundary method is one of the most useful computational methods in studying fluid structure interaction. On the other hand, the Immersed Boundary method is also known to have some limitations to limit its application. One of them is that it require small time steps to maintain stability when solved with an explicit method. If we want to study the long time evolution, the computational cost will be too expensive to afford. In order to overcome this difficulty, many implicit or approximately implicit methods have been proposed in the literature to remove this severe time step stability constraint, but none of them give satisfactory performance. In this paper, we propose some efficient semi-implicit schemes to remove this stiffness from the Immersed Boundary method for the steady Stokes, unsteady Stokes and Navier-Stokes equations respectively. These semi-implicit schemes have much better stability than the explicit schemes and their computational costs of each step are comparable to those of the explicit schemes. The construction of our semi-implicit scheme consists of two steps. First, we obtain a semi-implicit discretization which is proved to be unconditionally stable. This unconditionally stable semi-implicit scheme is still quite expensive to implement in practice. Next, we apply the Small Scale Decomposition to the unconditionally stable semi-implicit scheme to construct our efficient semi-implicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit scheme 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. Our extensive numerical experiments show that our semi-implicit scheme has much better stability property than an explicit scheme. This offers a substantial computational saving in using the Immersed Boundary method.