We consider a class of semi-Lagrangian high-order approximation schemes for convex Hamilton-Jacobi equations. In this framework, we prove that under certain restrictions on the relationship between $Delta x$ and $Delta t$, the sequence of approximate solutions is uniformly Lipschitz continuous and hence, by consistency, that it converges to the exact solution. The argument is suitable for most reconstructions of interest, including high-order polynomials and ENO reconstructions.
Ferretti, R. (2003). Convergence of semi-Lagrangian approximations to convex Hamilton-Jacobi equations under (very) large Courant numbers. SIAM JOURNAL ON NUMERICAL ANALYSIS, 40, 2240-2253.