Fourth-order finite difference scheme for the incompressible Navier-Stokes equations in a disk

We develop an efficient fourth-order finite difference method for solving the incompressible Navier-Stokes equations in the vorticitystream function formulation on a disk. We use the fourth-order Runge-Kutta method for the time integration and treat both the convection and diffusion terms explicitly. Using a uniform grid with shifting a half mesh away from the origin, we avoid placing the grid point directly at the origin; thus, no pole approximation is needed. Besides, on such grid, a fourth-order fast direct method is used to solve the Poisson equation of the stream function. By Fourier filtering the vorticity in the azimuthal direction at each time stage, we are able to increase the time step to a reasonable size. The numerical results of the accuracy test and the simulation of a vortex dipole colliding with circular wall are presented.