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

TL;DR

This paper introduces a new high-order shock capturing scheme for discontinuous finite element methods that preserves conservation and invariance, effectively resolving discontinuities in hyperbolic systems without tuning parameters.

Contribution

It generalizes the Lax-Friedrichs flux to a high-order staggered grid, providing a structure-preserving, parameter-free numerical scheme for nonlinear hyperbolic systems.

Findings

Successfully resolves discontinuities in Euler equations

Maintains conservation and invariance properties

Reduces spurious oscillations and dissipation

Abstract

We present a novel structure-preserving numerical scheme for discontinuous finite element approximations of nonlinear hyperbolic systems. The method can be understood as a generalization of the Lax-Friedrichs flux to a high-order staggered grid and does not depend on any tunable parameters. Under a presented set of conditions, we show that the method is conservative and invariant domain preserving. Numerical experiments on the Euler equations show the ability of the scheme to resolve discontinuities without introducing excessive spurious oscillations or dissipation.