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We develop an immersed boundary (IB) method for modeling flows around fixedor moving rigid bodies that is suitable for a broad range of Reynolds numbers,including steady Stokes flow. The spatio-temporal discretization of the fluidequations is based on a standard staggered-grid approach. Fluid-bodyinteraction is handled using Peskin's IB method; however, unlike existing IBapproaches to such problems, we do not rely on penalty or fractional-stepformulations. Instead, we use an unsplit scheme that ensures the no-slipconstraint is enforced exactly in terms of the Lagrangian velocity fieldevaluated at the IB markers. Fractional-step approaches, by contrast, canimpose such constraints only approximately. Imposing these constraints exactlyrequires the solution of a large linear system that includes the fluid velocityand pressure as well as Lagrange multiplier forces that impose the motion ofthe body. To solve this system efficiently, we develop a preconditioner for theconstrained IB formulation that is based on an analytical approximation to theSchur complement. This approach is enabled by the near translational androtational invariance of Peskin's IB method. We demonstrate that only a fewcycles of a geometric multigrid method for the fluid equations are required ineach application of the preconditioner, and we demonstrate robust convergenceof the overall Krylov solver despite the approximations made in thepreconditioner. We apply the method to a number of test problems at zero andfinite Reynolds numbers, and we demonstrate first-order convergence of themethod to several analytical solutions and benchmark computations.

An Immersed Boundary Method for Rigid Bodies