Spatial Approximation for Evolutionary Equations

TL;DR

This paper develops a general theoretical framework for approximating evolutionary equations, providing a foundation for numerical analysis, and demonstrates its application through spectral spatial discretization of the heat equation.

Contribution

It introduces a unified approximation theory for evolutionary equations, bridging the gap between abstract theory and practical numerical methods.

Findings

Established a general approximation framework for evolutionary equations

Applied the theory to spectral spatial discretization of the heat equation

Validated the approach through concrete numerical examples

Abstract

We consider evolutionary equations as introduced by R.\ Picard in 2009 and develop a general theory for approximation which can be seen as a theoretical foundation for numerical analysis for evolutionary equations. To demonstrate the approximation result, we apply it to a spatial discretisation of the heat equation using spectral methods.