We consider in this paper numerical approximation of the linear Fluid-Structure Interaction (FSI). We construct a new class of pressure-correction schemes for the linear FSI problem with a fixed interface, and prove rigorously that they are unconditionally stable. These schemes are computationally very efficient, as they lead to, at each time step, a coupled linear elliptic system for the velocity and displacement in the whole region and a discrete Poisson equation in the fluid region.

The rotational incremental pressure-correction (RIPC) scheme, described in Timmermans et al. [Int. J. Numer. Methods. Fluids., 22 (1996)] and Shen et al. [Math. Comput., 73 (2003)] for non-rotational Navier-Stokes equations, is extended to rotating incompressible flows. The method is implemented in the context of a pseudo Fourier-spectral code and applied to several rotating laminar and turbulent flows. The performance of the scheme and the computational results are compared to the so-called diagonalization method (DM) developed by Morinishi et al. [Int. J. Heat. Fluid. Flow., 22 (2001)]. The RIPC predictions are in excellent agreement with the DM predictions, while being simpler to implement and computationally more efficient. The RIPC scheme is not in anyway limited to implementation in a pseudo-spectral code or periodic boundary conditions, and can be used in complex geometries and with other suitable boundary conditions.