Optimal vorticity accuracy in an efficient velocity-vorticity method for the 2D Navier-Stokes equations

We study a velocity-vorticity scheme for the 2D incompressible Navier-Stokes equations, which is based on a formulation that couples the rotation form of the momentum equation with the vorticity equation, and a temporal discretization that stably decouples the system at each time step and allows for simultaneous solving of the vorticity equation and velocity-pressure system (thus if special care is taken in its implementation, the method can have no extra cost compared to common velocity-pressure schemes). This scheme was recently shown to be unconditionally long-time H-1 stable for both velocity and vorticity, which is a property not shared by any common velocity-pressure method. Herein, we analyze the scheme's convergence, and prove that it yields unconditional optimal accuracy for both velocity and vorticity, thus making it advantageous over common velocity-pressure schemes if the vorticity variable is of interest. Numerical experiments are given that illustrate the theory and demonstrate the scheme's usefulness on some benchmark problems.