Runge-Kutta and Networks

TL;DR

This paper categorifies Runge-Kutta methods by demonstrating that affine transformations of ODEs induce related discrete dynamical systems, with applications to coupled cell networks and fibrations of manifolds.

Contribution

It introduces a categorification of Runge-Kutta methods, linking affine transformations of ODEs to related discrete systems, expanding the theoretical framework of numerical integration.

Findings

Affine maps relate ODEs and their discrete systems.

Categorification applies well to coupled cell networks.

The approach extends to fibrations of networked manifolds.

Abstract

We categorify the RK family of numerical integration methods (explicit and implicit). Namely we prove that if a pair of ODEs are related by an affine map then the corresponding discrete time dynamical systems are also related by the map. We show that in practice this works well when the pairs of related ODEs come from the coupled cell networks formalism and, more generally, from fibrations of networks of manifolds.