  In this work, the Immersed Boundary Method (IBM) with feedback forcing
introduced by Goldstein et al. (1993) and often referred in the literature as
the Virtual Boundary Method (VBM), is addressed. The VBM has been extensively
applied both within a Spectral and a Finite Difference (FD) framework. Here, we
propose to combine the VBM with a computationally efficient Finite Volume (FV)
method. We will show that for similar computational configurations, FV and FD
methods provide significantly different results. Furthermore, we propose to
modify the standard feedback forcing scheme, based on a Proportional-Integral
(PI) controller, with the introduction of a derivative action, in order to
obtain a Proportial-Integral-Derivative (PID) controller. The stability
analysis for the Backward Differentiation Formula of order 1 (BDF1) time scheme
is modified accordingly, and extended to the Backward Differentiation Formula
of order 2 (BDF2) time scheme. We will show that, for the BDF2 time scheme, the
derivative action allows to improve the stability characteristics of the
system. Our approach is validated against numerical data available in the
literature for a stationary/rigidly moving 2D circular cylinder in several
configurations. Finally, a Fluid-Structure Interaction (FSI) benchmark, related
to the frequency response of a cantilever beam coupled with a fluid, is
presented: we numerically demonstrate that the introduction of the derivative
action plays an important role in order to properly detect the fluid-structure
interaction coupling.
