On the relationship between Semi-Lagrangian and Lagrange-Galerkin schemes

Following a previous result stating their equivalence under constant advection speed, Semi-Lagrangian and Lagrange-Galerkin schemes are compared in this paper in the situation of variable coefficient advection equations. Once known that Semi-Lagrangian schemes can be proved to be equivalent to area-weighted Lagrange-Galerkin schemes via a suitable definition of the basis functions, we will further prove that area-weighted Lagrange-Galerkin schemes represent a ``small" (more precisely, an $O(\Delta t$)) perturbation of exact Lagrange-Galerkin schemes. This equivalence implies a general result of stability for Semi-Lagrangian schemes.