PDE approximation of large systems of differential equations

TL;DR

This paper develops PDE-based approximations for large systems of ODEs, providing estimates on their accuracy and demonstrating advantages in a voter model example.

Contribution

It introduces PDE approximations with boundary conditions for large ODE systems and offers theoretical estimates on their accuracy.

Findings

PDE approximations closely match large ODE systems

Fourier method enhances analysis of voter model

Theoretical bounds on approximation errors

Abstract

A large system of ordinary differential equations is approximated by a parabolic partial differential equation with dynamic boundary condition and a different one with Robin boundary condition. Using the theory of differential operators with Wentzell boundary conditions and similar theories, we give estimates on the order of approximation. The theory is demonstrated on a voter model where the Fourier method applied to the PDE is of great advantage.