Stability of some generalized Godunov schemes with linear high-order reconstructions

Classical Semi-Lagrangian schemes have the advantage of allowing large time steps, but fail in general to be conservative. Trying to keep the advantages of both large time steps and conservative structure, Flux-Form Semi-Lagrangian schemes have been proposed for various problems, in a form which represent (at least in a single space dimension) a high-order, large time-step generalization of the Godunov scheme. At the theoretical level, a recent result has shown the equivalence of the two versions of Semi-Lagrangian schemes for constant coefficient advection equations, while, on the other hand, classical Semi-Lagrangian schemes have been proved to be strictly related to area-weighted Lagrange-Galerkin schemes for both constant and variable coefficient equations. We address in this paper a further issue in this theoretical framework, i.e., the relationship between stability of classical and of Flux-Form Semi-Lagrangian schemes. We show that the stability of the former class implies the stability of the latter, at least in the case of the one-dimensional linear continuity equation.