Journal

(Submitted)

Abstract

We propose a fixed-point-based numerical framework for computing stationary states of nonlocal Fokker-Planck-type equations. Instead of discretising the differential operators directly, we reformulate the stationary problem as a nonlinear fixed-point map built from the original PDE and its nonlocal interaction terms, and solve the resulting finite-dimensional problem with a matrix-free Newton-Krylov method. We compare implementations using the analytic Frechet derivative of this map with a simple central-difference approximation. Because the method does not rely on time evolution, it is agnostic to dynamical stability and can detect both stable and unstable stationary states. Its accuracy is determined mainly by the numerical treatment of convolutions and quadrature, rather than by differentiation stencils. We apply the approach to three model problems with linear diffusion, use existing analytical results to verify the outputs, and reproduce known bifurcation diagrams, as well as new bifurcation behaviour not previously observed in this kind of problem.

Overview

This paper is about building a numerical method for stationary states that avoids some of the usual challenges of directly discretising the differential operator and then evolving the PDE in time. Instead, the stationary problem is rewritten as a fixed-point problem, and the numerics are built around that formulation. One of the nice consequences is that the method is not tied to dynamical stability, so it can detect unstable stationary states as well as stable ones.

A big part of the paper is showing that this viewpoint is not just abstractly appealing, but actually useful in practice. We test it on several different nonlocal Fokker-Planck-type models, compare analytic and finite-difference versions of the associated Fréchet derivatives, and check the output against known analytical results whenever possible. The method also reproduces known bifurcation diagrams and picks up some new branch behaviour that had not been reported before.

In a loose sense, the paper is trying to show that if your real interest is in stationary states, then it can be more natural to solve the stationarity condition directly through a consistency map than to approximate the full time-dependent dynamics and wait for the system to settle. That gives a fairly flexible numerical framework for exploring the stationary structure of these nonlocal problems.

Keywords

nonlocal Fokker-Planck equations; stationary states; fixed-point methods; Newton-Krylov methods; bifurcation diagrams; matrix-free numerics

MSC 2020

35Q84; 65H10; 49M15; 35R09

Links

• Preprint (arXiv)