I would like to write my own solver for compressible Euler equations, and most importantly I want it to work robustly in all situations. I would like it to be FE based (DG is ok). What are the possible methods?
I am aware of doing 0th order DG (finite volumes) and that should work very robustly. I have implemented a basic FVM solver and it works great, but the convergence is quite slow. However, this is definitely one option.
I have implemented an FE solver (works for any mesh and any polynomial order on any element) for linearized Euler equations, but I am getting spurious oscillations (and eventually it blows out, so I can't use it so solve my problem) and I have read in the literature that one needs to stabilize it. If I implement some stabilization, would that work robustly for all problems (=boundary conditions and geometries)? What will be the convergence rate?
Other than that, is there some other robust methodology for Euler equations (i.e. higher order DG with some stabilization)?
I am aware that many people tried lots of different things in their research codes, but I am interested in a robust method that works for all geometries and boundary conditions (edit: in 2D and 3D).