Summation by parts (SBP) operators and interface boundary treatments

Finished of Carpenter, Nordstrom and Gottlieb's paper, "A stable and conservative interface treatment of arbitrary spatial accuracy". I like papers like this. It 'feels' clean, organised self contained. Writing such papers takes work so when I find ones like this I take a bit more time over them.

The crux of the paper is a penalty term in the difference approximation to the time derivative to implement the interface. In principle this seems the same as the SAT boundary conditions (see Carpenter, Gottlieb and Ararbanel's paper, "Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems" - which is another pleasurable paper to read). Given the success of SAT it's no surprise that penalty boundary conditions perform better than some other standard interface treatments, particular for SBP operators.

In any case I'm most interested in the presentation of a specific 4th order SBP operator in the last section of the appendix. Note that they reference Gustafsson's paper to justify the loss of one order of accuracy close to the boundary. This makes me feel uneasy, but it seems to work.