A Comparison of Two Adaptive Versions of the Immersed Boundary Method

The immersed boundary (IB) method is an approach to problems of fluid-structure interaction in which an elastic structure is immersed in a viscous incompressible fluid. In the continuous setting, the IB method couples an Eulerian description of the fluid to a Lagrangian description of the elastic structure via integral equations with Dirac delta function kernels, and in the discrete case, the delta function is approximated by a regularized version of the delta function. To improve the efficiency of the IB method, at least two different adaptive versions of the conventional IB methodology have been developed, one based on a staggered-grid discretization of the incompressible Navier-Stokes equations [1, 2] and the other based on a cellcentered discretization [3, 4], but we are aware of no direct comparisons between these two adaptive numerical methods. In this work, we present numerical experiments which allow us to compare directly the performance of adaptive cell-centered and staggered-grid IB methods. Our results indicate that the staggered-grid discretization generally yields reduced grid dependence, especially when the scheme is equipped with versions of the regularized delta function which satisfy a so-called even-odd condition. In our numerical experiments, we also find that the staggeredgrid scheme is dramatically more accurate than the cell-centered discretization in terms of volume conservation.