Order Stars and a Saturation Theorem for First-order Hyperbolics

We prove that stable numerical finite difference methods forfirst-order hyperbolics, which use s forward and r backwardsteps in the discretization of the space derivatives, are oforder at most 2 min{r+1, s}. This generalizes results of Strang(1964) and of Engquist & Osher (1980b). We also derive linearstability results for interpolatory finite differences. Thegiven analysis is based on a generalization of the theory oforder stars.