Approximating semigroups by using pseudospectra

TL;DR

This paper introduces a pseudospectra-based method for approximating solutions to evolution equations with non-self-adjoint generators, improving conditioning over traditional spectral expansions.

Contribution

It presents a novel pseudospectra approach for spectral approximation of non-self-adjoint operators, with theoretical insights and numerical validation.

Findings

Pseudospectra-based method is better conditioned than spectral expansion

The approach effectively approximates evolution equations with non-self-adjoint generators

Numerical examples demonstrate improved stability and accuracy

Abstract

We study evolution equations with non-self-adjoint generators, for example the convection-diffusion equation. Spectral expansions are not a reliable method of solving such equations, because they are so ill-conditioned. We introduce a new method using pseusospectra to produce an approximated spectral expansion, and explain its theoretical status. We also give some simple numerical examples to show that it is much better conditioned.