Discretization of the wave equation on a metric graph

TL;DR

This paper investigates how to discretize the wave equation on a metric graph, deriving node conditions via variational methods and exploring boundary controllability, bridging discrete and continuous cases.

Contribution

It introduces a variational approach to determine node conditions for the wave equation on discrete graphs, linking discrete and continuous frameworks.

Findings

Derived node conditions for wave equations on graphs

Established parallels between discrete and continuous cases

Analyzed boundary controllability on metric graphs

Abstract

The question of what conditions should be set at the nodes of a discrete graph for the wave equation with discrete time is investigated. The variational method for the derivation of these conditions is used. A parallel with the continuous case is also drawn. As an example the problem of shape controllability from the boundary is studied.