Splitting methods for the nonlocal Fowler equation

TL;DR

This paper develops and analyzes a splitting method for a nonlocal scalar conservation law modeling dune dynamics, proving convergence and demonstrating effectiveness through numerical experiments.

Contribution

It introduces a splitting-based numerical scheme for the Fowler equation and proves its convergence, combining Fourier and finite difference methods.

Findings

Proved convergence of the Lie splitting method for the Fowler equation.

Validated the numerical scheme through experiments confirming theoretical results.

Demonstrated efficiency of the split-step Fourier method in approximating solutions.

Abstract

We consider a nonlocal scalar conservation law proposed by Andrew C. Fowler to describe the dynamics of dunes, and we develop a numerical procedure based on splitting methods to approximate its solutions. We begin by proving the convergence of the well-known Lie formula, which is an approximation of the exact solution of order one in time. We next use the split-step Fourier method to approximate the continuous problem using the fast Fourier transform and the finite difference method. Our numerical experiments confirm the theoretical results.