A convergent finite volume method for the Kuramoto equation and related non-local conservation laws

TL;DR

This paper introduces a convergent finite volume method for the Kuramoto equation and similar nonlocal conservation laws, proving convergence and demonstrating effectiveness through numerical examples.

Contribution

It develops a Lax--Friedrichs type finite volume scheme with proven convergence properties for nonlocal continuity equations in multiple dimensions.

Findings

Method converges weakly to measure-valued solutions.

Strong convergence for initial data of bounded variation.

Numerical tests show effectiveness for regular and singular data.

Abstract

We derive and study a Lax--Friedrichs type finite volume method for a large class of nonlocal continuity equations in multiple dimensions. We prove that the method converges weakly to the measure-valued solution, and converges strongly if the initial data is of bounded variation. Several numerical examples for the kinetic Kuramoto equation are provided, demonstrating that the method works well both for regular and singular data.