Quadrature-free DG method using nodal Lagrangian basis are computationally very efficient. I have seen many papers using this method for linear PDE but almost no literature for non-linear PDE like compressible Euler equations. The book of Hesthaven & Warburton discusses this and talks of errors in quadrature-free method for non-linear problems and the need to use some filter. The few papers I have seen on quadrature-free method like this one use the same degree basis for solution and flux and dont seem to use any filter (but they have artificial viscosity which is probably enough). I would like to ask the experts if there are any issues in using quadrature-free approach for non-linear problems when I am already using a non-linear limiter or artificial viscosity ? Do they give the same convergence rates as the quadrature-based methods ?
I think this paper by H.L. Atkins and C.-W. Shu is exactly what you are looking for: