High-order finite difference schemes for incompressible flows

This paper presents a new high-order approach to the numerical solution of the incompressible Stokes and Navier-Stokes equations. The class of schemes developed is based upon a velocity-pressure-pressure gradient formulation, which allows: (i) high-order finite difference stencils to be applied on non-staggered grids; (ii) high-order pressure gradient approximations to be made using standard Pade schemes, and (iii) a variety of boundary conditions to be incorporated in a natural manner. Results are presented in detail for a selection of two-dimensional steady-state test problems, using the fourth-order scheme to demonstrate the accuracy and the robustness of the proposed methods. Furthermore, extensions to higher orders and time-dependent problems are illustrated, whereas the extension to three-dimensional problems is also discussed. Copyright (C) 2010 John Wiley & Sons, Ltd.