A Numerical Method for a Nonlocal Diffusion Equation with Additive Noise

TL;DR

This paper develops and analyzes numerical schemes for a nonlocal stochastic diffusion equation on graphons, providing convergence rates and justifying the continuum limit for noisy interacting particle systems.

Contribution

It introduces a well-posedness proof, semidiscrete and fully discrete schemes, and convergence analysis for a nonlocal stochastic PDE on graphons, including models with low regularity.

Findings

Semidiscrete scheme converges with rates depending on graphon regularity.

Fully discrete Euler-Maruyama scheme shows higher convergence speed in some models.

Numerical experiments confirm theoretical convergence rates.

Abstract

We consider a nonlocal evolution equation representing the continuum limit of a large ensemble of interacting particles on graphs forced by noise. The two principle ingredients of the continuum model are a nonlocal term and Q-Wiener process describing the interactions among the particles in the network and stochastic forcing respectively. The network connectivity is given by a square integrable function called a graphon. We prove that the initial value problem for the continuum model is well-posed. Further, we construct a semidiscrete (discrete in space and continuous in time) and a fully discrete schemes for the nonlocal model. The former is obtained by a discontinuous Galerkin method and the latter is based on further discretizing time using the Euler-Maruyama method. We prove convergence and estimate the rate of convergence in each case. For the semidiscrete scheme, the rate of convergence estimate is expressed in terms of the regularity of the graphon, Q-Wiener process, and the initial data. We work in generalized Lipschitz spaces, which allows to treat models with data of lower regularity. This is important for applications as many interesting types of connectivity including small-world and power-law are expressed by graphons that are not smooth. The error analysis of the fully discrete scheme, on the other hand, reveals that for some models common in applied science, one has a higher speed of convergence than that predicted by the standard estimates for the Euler-Maruyama method. The rate of convergence analysis is supplemented with detailed numerical experiments, which are consistent with our analytical results. As a by-product, this work presents a rigorous justification for taking continuum limit for a large class of interacting dynamical systems on graphs subject to noise.