  We demonstrate the applicability and advantages of Discontinuous Galerkin
(DG) schemes in the context of the Functional Renormalization Group (FRG). We
investigate the $O(N)$-model in the large $N$ limit. It is shown that the flow
equation for the effective potential can be cast into a conservative form. We
discuss results for the Riemann problem, as well as initial conditions leading
to a first and second order phase transition. In particular, we unravel the
mechanism underlying first order phase transitions, based on the formation of a
shock in the derivative of the effective potential.
