Atm it seems that we can stably, and also fairly good, evolve a stable TOV in Cowling approximation with resolution that is comparable (in one dimension) with that of Deppe's paper,
that is, n = 5, k = 16-24.
What is interesting is that it seems like we don't need an AV for the stable star.
The key change seems to have been the 'freezing' of the atmosphere, combined with the velocity limiter from the Dumbser paper.
Freezing atmosphere means that we restrict the points at which we are allowed to impose the atmosphere within ERK substeps.
Which points are allowed to evolve during substeps is determined from the initial data of each step.
The idea is to enforce a finite propagation speed for the atmosphere treatment.
Whether that works also in dynamical scenarios needs to be checked.
With regards to the finite propagation speed. I think one can cook up a nice argument in relation to a CFL like condition to show that freezing the atmosphere makes sense (=^= light cones).
