A second-order Magnus-type integrator for evolution equations with delay

TL;DR

This paper develops a second-order Magnus-type integrator for delay evolution equations, proving its convergence and invariance properties, with an application to a space-dependent epidemic model with latent period.

Contribution

It introduces a novel Magnus-type integrator for delay equations and establishes its convergence and invariance properties under weaker assumptions.

Findings

Proves second-order convergence of the integrator.

Shows the integrator preserves invariant sets under certain conditions.

Demonstrates applicability to a space-dependent epidemic model.

Abstract

We rewrite abstract delay equations to nonautonomous abstract Cauchy problems allowing us to introduce a Magnus-type integrator for the former. We prove the second-order convergence of the obtained Magnus-type integrator. We also show that if the differential operators involved admit a common invariant set for their generated semigroups, then the Magnus-type integrator will respect this invariant set as well, allowing for much weaker assumptions to obtain the desired convergence. As an illustrative example we consider a space-dependent epidemic model with latent period and diffusion.