A stable semi-discrete central scheme for the two-dimensional incompressible Euler equations

We derive a second-order, semi-discrete central-upwind scheme for the incompressible 2D Euler equations in the vorticity formulation. The reconstructed velocity field preserves an exact discrete incompressibility relation. We state a local maximum principle for a fully discrete version of the scheme and prove it using a convexity argument. We then show how similar convexity arguments can be used to prove that the scheme maps certain Orlicz spaces into themselves. The consequences of this result on the convergence of the scheme are discussed. Numerical simulations support the expected properties of the scheme.