A high-order accurate method for the numerical solution of the incompressible Navier-Stokes equations is developed. Fourth-order, explicit, finite difference schemes on staggered grids are used for space discretization. The explicit, fourth-order Runge-Kutta method is used for time marching. Incompressibility is enforce for each Runge-Kutta stage either by a local pressure correction, which is carried out for each cell using a local, fourth-order discrete analog of the continuity equation, or

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... solving numerically a global Poissontype equation also discretized to fourth-order accuracy and solved using GMRES. In both cases, the updated pressure is used to recompute the velocities until the incompressibility constraint is satisfied. The accuracy and efficiency of the proposed method is demonstrated for test problems.