A note on the use of central schemes for incompressible Navier–Stokes flows

Recently, central methods combined with ENO limiting [1–3] have become very popular for hyperbolic problems. The main advantage is the simplicity of this method since no Riemann problem has to be solved. The only information necessary to know from the system under consideration is an estimate of the spectral radius of the linearization of the flux, corresponding to the maximum wave speeds of the underlying system. Therefore, the method is attractive also for problems where the Riemann or approximate Riemann problem is too difficult to solve or to implement. This method has also been applied to the incompressible Navier–Stokes equation in two dimensions using the vorticity-stream function approach [3,4]. This approach has been questioned by Nielsen and Naulin [5]. They compared the CWENO-scheme as introduced by Kurganov and Levy (section 5, example 5) [3] with a standard spectral scheme and a finite difference approach using the Arakawa [6] discretization. The comparison focused on the conservation of integral quantities as the total energy and the total enstrophy. Their conclusion was that the spectral and Arakawa scheme outperformed the CWENO scheme quite dramatically in respect of numerical dissipation. The result is that both the Arakawa and the spectral scheme converge to the true solution from above in respect to the global quantities (energy and enstrophy are too high if underresolved) whereas the CWENO scheme converges (more slowly) from below (energy and enstrophy are too low). Nielsen and Naulin [5] did not compare the amount of spurious oscillations where the CWENO proves to have much better properties. Here, we demonstrate that simply switching from the stream-function approach to the integration of the primitive variables u with a projection method as discussed by Brown et al. [9] reduces the numerical dissipation quite substantially so that the CWENO scheme approaches the properties of the spectral and the Arakawa scheme without producing oscillations near strong vortex sheets. For a second order scheme