Newton's method for high-order schemes: as good as it gets?

High order Discontinuous Galerkin discretization schemes are considered for steady state problems. We discuss the issue of oscillations arising when Newton's method is employed to obtain a steady state solution. It will be demonstrated that flux approximation near flux extrema may produce spurious oscillations propagating over the domain of computation. The control over the numerical flux in the problem allows us to obtain non-oscillating convergent solutions.