Stability conditions for the numerical solution of convection-dominated problems with skew-symmetric discretizations

This paper presents original and close to optimal stability conditions linking the time step and the space step, stronger than the CFL criterion: $\delta t\le C\delta x^{\alpha }$ with $\alpha =\frac{2r}{2r-1}$, $r$ an integer, for some numerical schemes we produce, when solving convection-dominated problems. We test this condition numerically and prove that it applies to nonlinear equations under smoothness assumptions.