On the instability of a nonlocal conservation law

TL;DR

This paper investigates the stability of a nonlocal conservation law modeling sand dune morphodynamics, proving that constant solutions are non-linearly unstable through analytical and numerical methods.

Contribution

It establishes the non-linear instability of constant solutions in Fowler's nonlocal conservation law, combining theoretical proof with numerical illustration.

Findings

Constant solutions are non-linearly unstable.

Numerical simulations confirm analytical instability.

The study advances understanding of dune morphodynamics modeling.

Abstract

We are interested in a nonlocal conservation law which describes the morphodynamics of sand dunes sheared by a fluid flow, recently proposed by Andrew C. Fowler. We prove that constant solutions of Fowler's equation are non-linearly unstable. We also illustrate this fact using a finite difference scheme.