A convergent finite volume method for the Kuramoto equation and related non-local conservation laws
Provides a convergent numerical scheme for a class of nonlocal PDEs, but the method is an adaptation of existing techniques and the results are incremental for the PDE community.
The authors developed a Lax-Friedrichs finite volume method for nonlocal continuity equations, proving weak convergence to measure-valued solutions and strong convergence for BV initial data. Numerical tests on the Kuramoto equation show effectiveness for regular and singular data.
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.