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Common efficient schemes for the incompressible Navier-Stokes equations, suchas projection or fractional step methods, have limited temporal accuracy as aresult of matrix splitting errors, or introduce errors near the domainboundaries (which destroy uniform convergence to the solution). In this paperwe recast the incompressible (constant density) Navier-Stokes equations (withthe velocity prescribed at the boundary) as an equivalent system, for theprimary variables velocity and pressure. We do this in the usual way away fromthe boundaries, by replacing the incompressibility condition on the velocity bya Poisson equation for the pressure. The key difference from the usualapproaches occurs at the boundaries, where we use boundary conditions thatunequivocally allow the pressure to be recovered from knowledge of the velocityat any fixed time. This avoids the common difficulty of an, apparently,over-determined Poisson problem. Since in this alternative formulation thepressure can be accurately and efficiently recovered from the velocity, therecast equations are ideal for numerical marching methods. The new system canbe discretized using a variety of methods, in principle to any desired order ofaccuracy. In this work we illustrate the approach with a 2-D second orderfinite difference scheme on a Cartesian grid, and devise an algorithm to solvethe equations on domains with curved (non-conforming) boundaries, including acase with a non-trivial topology (a circular obstruction inside the domain).This algorithm achieves second order accuracy (in L-infinity), for both thevelocity and the pressure. The scheme has a natural extension to 3-D.

journal of computational physics

An efficient method for the incompressible Navier-Stokes equations on  irregular domains with no-slip boundary conditions, high order up to the  boundary