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We present a novel structure-preserving numerical scheme for discontinuousfinite element approximations of nonlinear hyperbolic systems. The method canbe understood as a generalization of the Lax-Friedrichs flux to a high-orderstaggered grid and does not depend on any tunable parameters. Under a presentedset of conditions, we show that the method is conservative and invariant domainpreserving. Numerical experiments on the Euler equations show the ability ofthe scheme to resolve discontinuities without introducing excessive spuriousoscillations or dissipation.

A Riemann Difference Scheme for Shock Capturing in Discontinuous Finite Element Methods