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 fourth-order 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.