Hydrodynamics of a Discrete Conservation Law

TL;DR

This paper classifies and explores complex hydrodynamic solutions of a discrete conservation law, revealing novel wave phenomena and solution types beyond classical continuum analogs through theoretical and numerical analysis.

Contribution

It introduces a detailed classification of solutions for a discrete conservation law, uncovering new wave behaviors and solution families not present in continuum models.

Findings

Discovery of discrete analogues of dispersive shock waves and rarefaction waves.

Identification of new solution families with no continuum counterparts.

Observation of finite time blow-up and complex wave interactions.

Abstract

The Riemann problem for the discrete conservation law $2 \dot{u}_n + u^2_{n+1} - u^2_{n-1} = 0$ is classified using Whitham modulation theory, a quasi-continuum approximation, and numerical simulations. A surprisingly elaborate set of solutions to this simple discrete regularization of the inviscid Burgers' equation is obtained. In addition to discrete analogues of well-known dispersive hydrodynamic solutions -- rarefaction waves (RWs) and dispersive shock waves (DSWs) -- additional unsteady solution families and finite time blow-up are observed. Two solution types exhibit no known conservative continuum correlates: (i) a counterpropagating DSW and RW solution separated by a symmetric, stationary shock and (ii) an unsteady shock emitting two counter-propagating periodic wavetrains with the same frequency connected to a partial DSW or a RW. Another class of solutions called traveling DSWs, (iii), consists of a partial DSW connected to a traveling wave comprised of a periodic wavetrain with a rapid transition to a constant. Portions of solutions (ii) and (iii) are interpreted as shock solutions of the Whitham modulation equations.