  A computationally efficient method for solving three-dimensional, viscous,
incompressible flows on unbounded domains is presented. The method formally
discretizes the incompressible Navier-Stokes equations on an unbounded
staggered Cartesian grid. Operations are limited to a finite computational
domain through a lattice Green's function technique. This technique obtains
solutions to inhomogeneous difference equations through the discrete
convolution of source terms with the fundamental solutions of the discrete
operators. The differential algebraic equations describing the temporal
evolution of the discrete momentum equation and incompressibility constraint
are numerically solved by combining an integrating factor technique for the
viscous term and a half-explicit Runge-Kutta scheme for the convective term. A
projection method that exploits the mimetic and commutativity properties of the
discrete operators is used to efficiently solve the system of equations that
arises in each stage of the time integration scheme. Linear complexity, fast
computation rates, and parallel scalability are achieved using recently
developed fast multipole methods for difference equations. The accuracy and
physical fidelity of solutions is verified through numerical simulations of
vortex rings.
