  This paper presents original and close to optimal stability conditions
linking the time step and the space step, stronger than the CFL criterion:
$\delta t\leq C\delta x^\alpha$ 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.
