Some recent results on the convergence of Semi-Lagrangian schemes

We review some recent constributions to the theory of convergence for SL schemes. A first result concerns the equivalence between SL and exact Lagrange-Galerkin schemes, which allows to recover $L2$ stability of SL schemes in a variety of cases, under both small and large Courant numbers. A second result is related to an adaptation of the classical Barles-Souganidis theorem for the approximation of viscosity solutions. In this situation we show that, under an inverse Courant condition, SL schemes behave like "asymptotically monotone" schemes, and this implies convergence to continuous viscosolutions.