Global Error Analysis and Inertial Manifold Reduction

TL;DR

This paper introduces a unified framework for analyzing four types of global errors in initial value problems, combining error amplification analysis with inertial manifold reduction techniques, demonstrated on complex dynamical systems.

Contribution

It presents a novel combined approach for global error analysis and inertial manifold reduction, extending classical methods to include time rescaling and applying them to chaotic systems.

Findings

Effective error bounds for Lorenz equations

Demonstrated dimension reduction for Kuramoto-Sivashinsky equation

Unified framework enhances understanding of error propagation

Abstract

Four types of global error for initial value problems are considered in a common framework. They include classical forward error analysis and shadowing error analysis together with extensions of both to rescaling of time. To determine the amplification of the local error that bounds the global error we present a linear analysis similar in spirit to condition number estimation for linear systems of equations. We combine these ideas with techniques for dimension reduction of differential equations via a boundary value formulation of numerical inertial manifold reduction. These global error concepts are exercised to illustrate their utility on the Lorenz equations and inertial manifold reductions of the Kuramoto-Sivashinsky equation.