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

Resolving phase transitions with Discontinuous Galerkin methods