Discretization of the wave equation on a metric graph Sergei A. Avdonin1 Aleksander S. Mikhaylov23 Victor S. Mikhaylov2. Abdon E. Choque-Rivero4 1Department of Mathematics Statistics University of Alaska Fairbanks Fairbanks Alaska USA

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Discretization of the wave equation on a metric graph

Sergei A. Avdonin1, Aleksander S. Mikhaylov2,3, Victor S. Mikhaylov2. Abdon E. Choque-Rivero4,

1Department of Mathematics & Statistics, University of Alaska Fairbanks, Fairbanks, Alaska, USA;

2St.Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences, St. Petersburg,

Russia; 3Department of Mathematics, St. Petersburg State University, St. Petersburg, Russia; 4Institute of

Physics and Mathematics, Universidad Michoacana de San Nicolas de Hidalgo, Michoacan, Mexico

ARTICLE HISTORY

Compiled Tuesday 18th October, 2022

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.

KEYWORDS

Discrete graph, node conditions, wave equation.

AMS CLASSIFICATION

35L05, 35Q93, 93B05, 93C20, 37M15, 37J51

1. Introduction

Under differential equation networks (DENs) or, in other words, quantum graphs we understand dif-

ferential equations on metric graphs coupled by certain vertex matching conditions. These models

play a fundamental role in many problems of science and engineering. The range for applications of

DENs is enormous and continues to grow; we will mention a few of them.

–Structural Health Monitoring. DENs, classically, arise in the study of stability, health, and oscilla-

tions of ﬂexible structures that are made of strings, beams, cables, and struts [1–4]

–Water, Electricity, Gas, and Trafﬁc Networks. An important example of DENs is the Saint-Venant

system of equations, which model hydraulic networks for water supply and irrigation [5]and ﬁrst-order

hyperbolic equations [6–11]and the isothermal Euler equations for describing the gas ﬂow through

pipelines [12,13].Other important examples of DENs include the telegrapher equation for modeling

electric networks [14], the diffusion equations in power networks [15], and Aw-Rascle equations for

describing road trafﬁc dynamics [16], see also [17]for trafﬁc ﬂow on networks and [18–20]for modeling

groundwater ﬂow.

–Nanoelectronics and Quantum Computing. Mesoscopic quasi-one-dimensional structures such as

quantum, atomic, and molecular wires are the subject of extensive experimental and theoretical stud-

ies [21,22]The simplest model describing conduction in quantum wires is the Schrödinger operator

on a planar graph. For similar models appear in nanoelectronics, high-temperature superconductors,

quantum computing and studies of quantum chaos see [23–25].

CONTACT Sergei A. Avdonin. Email: saavdonin@alaska.edu

arXiv:2210.09274v1 [math.OC] 17 Oct 2022

–Material Science. Quantum graphs arise in analyzing hierarchical materials like ceramic and

metallic foams, percolation networks, carbon and graphene nano-tubes, and graphene ribbons [26,

27].

–Biology. Challenging problems involving ordinary and partial differential equations on graphs

arise in signal propagation in dendritic trees, particle dispersal in respiratory systems, species per-

sistence and biochemical diffusion in delta river systems [28,29].

–Social Networks. Examples of social applications are modeling of international trades and space-

temporal patterns of information spread [30].

There are many papers in the literature devoted to study of the spectral properties of differential op-

erators on graphs and well-posedness of the initial boundary value problems for differential equations

on graphs and regularity of their solutions [31–37]. On the other hand, numerical methods for solving

ODEs and PDEs on graphs have been mostly developed only for very speciﬁc problems [11,38–44].

Only recently more general investigations appeared directed to developing the ﬁnite element method

for elliptic and parabolic equations on graphs [45–47].

In the present paper we study the problems of discretization of the wave equation on metric graphs.

The main attention is put to discretization of the Kirchoff – Neumann matching conditions at the in-

ternal vertices. This important problems has not got a proper attention in the literature. To derive the

matching conditions we apply Hamilton’s principle and variational methods in both continuous and

discrete cases. Then we discuss a proper choice of the nodal weights for the discrete model to insure

the same reﬂection and transmission coefﬁcients as the original continuous model has. The paper is

organized as follows. In Section 2 we derive the matching conditions for metric graphs and in Section

3 — for discrete graphs. In Section 4 we discuss transmission and reﬂection of the wave at the internal

nodes in both, continuous and discrete, situations. In Section 5 we solve direct and control problems

for the wave equation on a discrete star graph.

2. Conditions at nodes on a metric graph

Let Ωbe a ﬁnite connected compact graph. The graph consists of edges E={e1,...,eN}connected

at vertices V={v1...,vM}. Each edge ej∈Eis identiﬁed with an interval (0, lj)of the real line. The

boundary Γ={v1,...,vm}of Ωis the set of vertices whose degree is equal to one (outer nodes). Let

E(v)be a set of edges incident to v. In what follows, we assume that some of the boundary vertices are

clamped, and the non-homogeneous Dirichlet boundary condition is imposed on the other part.

The space of real square-integrable functions on the graph Ωis denoted by L2(Ω):=LN

i=1L2(ei). For

the function u∈L2(Ω)we will write

u:=uiN

i=1,ui∈L2(ei).

By Cwe denote the direct problem on the metric graph:

t t (x,t)−ui

x x (x,t) = 0x∈ei, (1)

ui(v,t) = uj(v,t),ei,ej∈E(v),v∈V\Γ, (2)

ei∈E(v)

∂ui(v,t) = 0, v∈V\Γ, (3)

u(v,t) = f(t)for v∈Γ, (4)

u(x,0) = 0, ut(x,0) = 0, for x∈Ω. (5)

Here ∂ui(v,t)is the derivative of uiat valong eiin the direction away from v. This system describes

small transverse oscillations of the graph (Figure 1), and u(x,t)denotes the displacement of the point

x∈Ωof the graph at time tfrom the equilibrium position. It is assumed that the boundary of the

graph moves according to the law f(t), and in the initial moment of time the graph was at rest in an

equilibrium state.

Figure 1. Wave equation on a metric graph with six edges and seven vertices. On the edge e1(resp. e4) the function f1(t)(resp. f4(t)) is

acting. On the edge e4the corresponding displacement u(x,t)is represented.

In order to write a similar system of equations for a discrete graph, let us recall how equations (1)–(5)

arose. We introduce the kinetic and potential energies:

T(t) = 1

2ZΩ

t(x,t)d x , (6)

U(t) = 1

2ZΩ

x(x,t)d x . (7)

According to the principles of classical mechanics (Hamilton’s principle of least action), the system

passes from state 1 at time t1to state 2 at time t2in such a way that the variation of the action functional

vanishes

S[u] = Zt2

L(t)d t ,L=T−U(Lagrangian), (8)

along the true trajectory of movement. It means that

δS[u,h] = 0 for all h:h|t=t1=0,h|t=t2=0, h|Γ=0.

Calculating the variation of the functional and integrating by parts, we get the following:

0=δS[u,h] = Zt2

t1ZΩ

(utht−uxhx)d x d t

=−Zt2

t1ZΩ

(ut t −ux x )h d x d t +Zt2

t1ZV\Γ

∂uh d x d t ,

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DiscretizationofthewaveequationonametricgraphSergeiA.Avdonin1,AleksanderS.Mikhaylov2,3,VictorS.Mikhaylov2.AbdonE.Choque-Rivero4,1DepartmentofMathematics&Statistics,UniversityofAlaskaFairbanks,Fairbanks,Alaska,USA;2St.PetersburgDepartmentofSteklovMathematicalInstitute,RussianAcademyofSciences,St.Pete...

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Discretization of the wave equation on a metric graph Sergei A. Avdonin1 Aleksander S. Mikhaylov23 Victor S. Mikhaylov2. Abdon E. Choque-Rivero4 1Department of Mathematics Statistics University of Alaska Fairbanks Fairbanks Alaska USA

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