  Dispersion of low-density rigid particles with complex geometries is
ubiquitous in both natural and industrial environments. We show that while
explicit methods for coupling the incompressible Navier-Stokes equations and
Newton's equations of motion are often sufficient to solve for the motion of
cylindrical particles with low density ratios, for more complex particles -
such as a body with a protrusion - they become unstable. We present an implicit
formulation of the coupling between rigid body dynamics and fluid dynamics
within the framework of the immersed boundary projection method. Similarly to
previous work on this method, the resulting matrix equation in the present
approach is solved using a block-LU decomposition. Each step of the block-LU
decomposition is modified to incorporate the rigid body dynamics. We show that
our method achieves second-order accuracy in space and first-order in time
(third-order for practical settings), only with a small additional
computational cost to the original method. Our implicit coupling yields stable
solution for density ratios as low as $10^{-4}$. We also consider the influence
of fictitious fluid located inside the rigid bodies on the accuracy and
stability of our method.
