Global Error Analysis and Inertial Manifold Reduction
This work provides a theoretical framework for error analysis in numerical solutions of differential equations, but is incremental as it extends existing concepts without introducing new paradigms or achieving SOTA results.
The paper unifies four types of global error analysis for initial value problems and combines them with inertial manifold reduction techniques, demonstrating their utility on Lorenz and Kuramoto-Sivashinsky equations.
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.