Matrix equation techniques for certain evolutionary partial differential equations

TL;DR

This paper introduces a new method for efficiently solving large-scale evolutionary partial differential equations by representing discretized operators as a Sylvester matrix equation and exploiting their structure.

Contribution

It presents a novel solution strategy combining projection techniques with entry-wise matrix structure to improve efficiency and reduce storage in solving large PDE discretizations.

Findings

Efficiently solves PDEs with many degrees of freedom.

Maintains low storage requirements.

Demonstrates effectiveness through numerical examples.

Abstract

We show that the discrete operator stemming from the time and space discretization of evolutionary partial differential equations can be represented in terms of a single Sylvester matrix equation. A novel solution strategy that combines projection techniques with the full exploitation of the entry-wise structure of the involved coefficient matrices is proposed. The resulting scheme is able to efficiently solve problems with a tremendous number of degrees of freedom while maintaining a low storage demand as illustrated in several numerical examples.