Riemann solvers and undercompressive shocks of convex FPU chains

TL;DR

This paper investigates how discrete FPU chains with convex potentials exhibit non-classical shocks, including undercompressive and dispersive shocks, which differ from classical p-system solutions, through systematic comparison and numerical analysis.

Contribution

It introduces modified p-system Riemann solvers to describe atomistic FPU chain solutions, revealing the existence of undercompressive shocks and their relation to flux convexity.

Findings

FPU chains can produce conservative shocks not predicted by classical p-systems.

Dispersive shocks replace Lax shocks in FPU chain solutions.

Convex-concave fluxes lead to supersonic conservative shocks, while concave-convex fluxes do not.

Abstract

We consider FPU-type atomic chains with general convex potentials. The naive continuum limit in the hyperbolic space-time scaling is the p-system of mass and momentum conservation. We systematically compare Riemann solutions to the p-system with numerical solutions to discrete Riemann problems in FPU chains, and argue that the latter can be described by modified p-system Riemann solvers. We allow the flux to have a turning point, and observe a third type of elementary wave (conservative shocks) in the atomistic simulations. These waves are heteroclinic travelling waves and correspond to non-classical, undercompressive shocks of the p-system. We analyse such shocks for fluxes with one or more turning points. Depending on the convexity properties of the flux we propose FPU-Riemann solvers. Our numerical simulations confirm that Lax-shocks are replaced by so called dispersive shocks. For convex-concave flux we provide numerical evidence that convex FPU chains follow the p-system in generating conservative shocks that are supersonic. For concave-convex flux, however, the conservative shocks of the p-system are subsonic and do not appear in FPU-Riemann solutions.