Splitting methods for the nonlocal Fowler equation
Provides a rigorous numerical framework for a specific nonlocal conservation law in geophysical modeling, but the approach is incremental.
The authors prove convergence of the Lie splitting method for the nonlocal Fowler equation describing dune dynamics, achieving first-order temporal accuracy, and validate it numerically using split-step Fourier and finite difference methods.
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.