Oscillatory Waves in Discrete Scalar Conservation Laws

TL;DR

This paper investigates Hamiltonian difference schemes for scalar conservation laws, establishing the existence of periodic travelling waves, analyzing their approximation, and providing numerical results to support the theoretical findings.

Contribution

It introduces a new framework for proving the existence of wavetrains in Hamiltonian schemes using integral equations and constrained maximization.

Findings

Existence of a three-parameter family of wavetrains

Development of an integral equation approach for wave profiles

Numerical results demonstrating wave approximations

Abstract

We study Hamiltonian difference schemes for scalar conservation laws with monotone flux function and establish the existence of a three-parameter family of periodic travelling waves (wavetrains). The proof is based on an integral equation for the dual wave profile and employs constrained maximization as well as the invariance properties of a gradient flow. We also discuss the approximation of wavetrains and present some numerical results.