I've just finished up my second reading of Gustafsson's, "The convergence rate for difference approximations to mixed initial boundary value problems" and I'm still abit confused.
The paper is referenced by (what seems to me) almost everyone and their dog as justification that a finite difference scheme can have one order of convergence less on boundary points than the design convergence rate and still achieve design. It strikes me that the paper works with multi-step schemes while it is sometimes referenced by papers using a semi-continuous or Runge-Kutta approach to time integration. This seems a little contradictory to me and while the bounds derived can probably be applied to these cases there does, in principle, seem to me a large difference between an operator used to estimate a derivative and an operator used to construct the next time level. Admittedly I'm not on top of the paper as it's quite dense and assumes knowledge of the equally dense, but far larger, paper by Gustafsson, Kreiss and Sundstrom, so there could be something I'm missing here. Perhaps it's "obvious" to the experts?
The main theorems, 2.1 and 2.2, both assume that every solution to the difference scheme is bounded by an appropriate norm applied to the initial conditions.
The techniques used to prove the theorems are quite nice, and follow the general structure used in Gustafsson, Kriess and Sundstrom: application of Fourier transforms and similarity transformations. Indeed compared to the difficult to swallow definitions the proofs don't take much work. It makes me reflect that in more modern papers I haven't seen much of this technique, perhaps there are better ways, or perhaps I'm looking in the wrong places.
Lastly, there is a nice example at the back of the paper which gives a much needed example.