$L^2$-stability of explicit schemes for incompressible Euler equations

We present an original study on the numerical stabiliy of explicit schemes solving the incompressible Euler equations on an open domain with slipping boundary conditions. Relying on the skewness property of the non-linear term, we demonstrate that some explicit schemes are numerically stable for small perturbations under the condition $\delta t\le C\delta x^{2r\mathrm{/}(2r-1)}$ where $r$ is an integer, $\delta t$ the time step and $\delta x$ the space step.