From the Newton equation to the wave equation : the case of shock waves

TL;DR

This paper investigates the transition from Newtonian particle dynamics to wave equations in the presence of shock waves, demonstrating that shocks cause deviations from smooth wave solutions in atomic chains.

Contribution

It provides a mathematical and numerical analysis showing shocks prevent the macroscopic limit from remaining a smooth wave equation in atomic chains.

Findings

Shocks occur when particle distances stay bounded.

The wave equation limit breaks down at shocks.

Numerical simulations confirm the theoretical predictions.

Abstract

We study the macroscopic limit of a chain of atoms governed by the Newton equation. It is known from the work of Blanc, Le Bris, Lions, that this limit is the solution of a nonlinear wave equation, as long as this solution remains smooth. We show, numerically and mathematically that, if the distances between particles remain bounded, it is not the case any more when there are shocks -at least for a convex nearest-neighbour interaction potential with convex derivative.