A staggered grid, high-order accurate method for the incompressible Navier-Stokes equations

A high-order accurate, finite-difference method for the numerical solution of the incompressible Navier–Stokes equations is presented. Fourth-order accurate discretizations of the convective and viscous fluxes are obtained on staggered meshes using explicit or compact finite-difference formulas. High-order accuracy in time is obtained by marching the solution with Runge–Kutta methods. The incompressibility constraint is enforced for each Runge–Kutta stage iteratively either by local pressure correction or by a Poisson-equation based global pressure correction method. Local pressure correction is carried out on cell by cell basis using a local, fourth-order accurate discrete analog of the continuity equation. The global pressure correction is based on the numerical solution of a Poisson-type equation which is discretized to fourthorder accuracy, and solved using GMRES. In both cases, the updated pressure is used to recompute the velocities in order to satisfy the incompressibility constraint to fourth-order accuracy. The accuracy and efficiency of the proposed method is demonstrated in test problems. 2005 Elsevier Inc. All rights reserved.