1 A Nonlinear Heat Equation Arising from Automated -Vehicle Traffic Flow Models

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A Nonlinear Heat Equation Arising from Automated-Vehicle

Traffic Flow Models

Dionysiοs Theodosis*, Iasson Karafyllis**, George Titakis*,

Ioannis Papamichail* and Markos Papageorgiou*,***

* Dynamic Systems and Simulation Laboratory,

Technical University of Crete, Chania, 73100, Greece,

emails: dtheodosis@dssl.tuc.gr, gtitakis@dssl.tuc.gr,

ipapa@dssl.tuc.gr, markos@dssl.tuc.gr

** Dept. of Mathematics, National Technical University of Athens,

Zografou Campus, 15780, Athens, Greece,

emails: iasonkar@central.ntua.gr , iasonkaraf@gmail.com

*** Faculty of Maritime and Transportation,

Ningbo University, Ningbo, China.

Abstract

In this paper, a new nonlinear heat equation is studied that arises as a

model of the collective behavior of automated vehicles. The properties of

the solutions of this equation are studied by introducing the appropriate

notion of a weak solution that requires certain entropy-like conditions. To

obtain an approximation of the solution of the nonlinear heat equation, a

new conservative first-order finite difference scheme is proposed that

respects the corresponding entropy conditions, and certain links between

the weak solution and the numerical scheme are provided. Finally, a traffic

simulation scenario and a comparison with the Lighthill-Witham-Richards

(LWR) model are provided, illustrating the benefits of the use of

automated vehicles.

Keywords: Nonlinear Heat Equation, Macroscopic Traffic Flow, Automated Vehicles

1. Introduction

The heat equation is a fundamental differential equation and among the most studied topics, see

for instance [11], [20], [21], [25], [26], [15], [27], [30], [33] and references therein. It describes the

evolution of the density of some quantity, such as temperature or chemical concentration, and finds

immediate application in a variety of topics such as financial mathematics, flow and pressure

diffusion in porous media, image analysis and many others.

In this paper, we consider a novel nonlinear heat equation, whose diffusion/viscosity coefficient

depends on both the density and the spatial derivative of the density (see also [11], [20], [21], [15]),

a feature that is rarely studied in the literature where typically the diffusion coefficient exclusively

depends on the density, see for instance [11], [25], [26], [27], [29], [30], [33] and references therein.

The nonlinear heat equation under consideration is inspired by a second order macroscopic traffic

flow model that was recently derived in [16] by the design of cruise controllers for autonomous

vehicles and has several fluid-like characteristics. Macroscopic traffic flow models have been widely

used as they are able to capture the collective behavior of vehicles in a traffic stream and can express

certain relationships between traffic flow, density and mean speed. Macroscopic traffic flow models

can be in general distinguished between first-order models, governed by the continuity equation, and

second-order models, which include an additional differential equation for the speed (see for instance

[2], [12], [34]).

In the present work, it is first shown that the solutions of the corresponding second-order

macroscopic model can be approximated by the solutions of this particular nonlinear heat equation.

Furthermore, inspired by the mechanical energy of the original second-order system, certain

functionals are defined, providing entropy-like conditions that characterize physically meaningful

solutions, see [9], [31]. In particular, we define the concept of a weak solution for the problem at

hand that is more demanding than the typical definition of weak solution (see [24], [31]), as it requires

that certain entropy conditions hold and, in addition, the mass is conserved. Finally, it is shown that,

under certain regularity assumptions, this type of weak solutions is equivalent to classical solutions,

under the assumption of certain entropy-like conditions (see Proposition 1).

The study of the nonlinear heat equation is inevitably related to numerical techniques in order to

obtain an approximation of its solution. More specifically, several finite-difference methods have

been proposed to study a variety of nonlinear heat equations, see for instance [3], [4], [6], [7], [8],

[10], [13], [14], and references therein. In this paper, we propose a new first-order explicit finite-

difference scheme that, in addition to being conservative (Proposition 2), it also respects the

corresponding entropy conditions (Proposition 3 and Proposition 5). Moreover, a condition on the

step size is provided that is a nonlinear version of the Courant-Friedrichs-Levy condition (Proposition

4). Finally, a link between the numerical solution produced by the proposed numerical scheme and

the corresponding weak solution is provided (Proposition 6). To demonstrate the properties of the

solutions of the nonlinear heat equation, we present two examples: (i) an academic example, where

the efficiency of the proposed numerical scheme is illustrated; and (ii) a realistic traffic simulation

scenario showing that the mean flow produced by automated vehicles is much higher than that of the

LWR (see [22], [28]).

Therefore, the contribution of the paper is summarized as follows:

• The study of a novel nonlinear heat equation that arises in traffic flow theory for automated

vehicles;

• The construction of a first-order numerical scheme that can be applied to nonlinear heat

equations with an explicit condition for numerical stability;

• The study of the connection of the weak solution for the nonlinear heat equation with the

solution produced by the proposed numerical scheme.

The structure of the paper is as follows. Section 2 is devoted to the presentation of a second-order

traffic flow model and the approximation of its solution by the solution of a particular nonlinear heat

equation. Section 3 presents the proposed numerical scheme and its properties. Section 4 presents the

two aforementioned examples. Finally, some concluding remarks are given in Section 5. All proofs

are provided in the Appendix.

Notation. Throughout this paper, we adopt the following notation.

||x

we denote both the Euclidean norm of a vector

x

and the absolute value of a scalar

x

:=[0, )

 +

denotes the set of non-negative real numbers.

x

we denote the transpose of a vector

x

. By

 

| | = max | |, =1,...,

x x i n



we denote the

infinity norm of a vector

= ( , ,..., ) n

x x x x 

Let

A

be an open set. By

0( ; )CA

, we denote the class of continuous functions on

A

, which take values in

  

. By

( ; )

CA

, where

1k

is an integer, we denote the class

of functions on

A

with continuous derivatives of order

, which take values in

  

When

=

the we write

0()CA

()

Let

I

be a given interval. For

[1, )p

()

denotes the set of equivalence classes of

Lebesgue measurable functions

:fI→

with

: | ( )|

f f x dx



=  +







()LI



denotes

the set of equivalence classes of measurable functions

:fI→

for which

( )

= ( ) <

sup

f f x

ess



+

Let

:u+

  → 

( , ) ( , )t x u t x→

be any function differentiable with respect to its arguments.

We use the notation

( , ) = ( , )

u t x t x



and

( , ) = ( , )

u t x t x



for the partial derivatives of

with

respect to

and

, respectively. We use the notation

[]ut

to denote the profile at certain

0t

( [ ])[ ]:= ( , )u t x u t x

, for all

x

For a set

S

denotes the closure of

2. A Nonlinear Heat Equation

2.1 Motivation and Derivation of a Nonlinear Heat Equation

In the recent paper [16], the study of microscopic vehicle movement control laws (cruise

controllers) for autonomous vehicles on lane-free roads brought forth the following macroscopic

model that holds for







( ) = 0v





(2.1)

( ) ( ) ( ) = ( ( ) '( ) ) ( )q v v q v vv P g v v f v v

    

      





+ + − −

(2.2)

with constraints



)

( , ) 0, max

   



( )

( , ) 0, max





for







, where

max





max

, and

max

(0, )vv



are constants. The states

( , )

  

( , )v



are the traffic density and

mean speed, respectively, at time



and position





on a highway, while the constants

max





max

, and

max

(0, )vv



are the maximum density, the maximum speed, and the

speed set-point, respectively. Moreover,

:f → 

is a

function with

(0) = 0f

and

( ) > 0f



for all





1()gC

is an increasing function with

'( ) > 0gv

for all

v

and

()qv

is defined by

222

( ) = 2( )

max max

max

v v vv v v

q v v v v v



−+

−

(2.3)

Model (2.1), (2.2) is a fluid-like model where the term

()P







is a pressure term and expresses

the tendency to accelerate or to decelerate based on the (local) density; while the term

( ( ) ( ) )g v v







, is a viscosity term, by analogy with the theory of fluids, with

()



playing the role

of dynamic viscosity, see [17], [23]. The functions

max

:[0, )



→



)

: 0, max



→

are



)

( )

10, max



and satisfy the following properties

( ) =

lim

max





−

→

+

(2.4)

( ) = 0, ( ) = 0P

  

for all

[0, ]





, and

( ) 0





( ) 0P





for all

max

( , )

  



(2.5)

where the constant

( )

0, max





is referred to as the interaction density due to the fact that for all

[0, ]





there is no interaction among the vehicles (see (2.5)). The term

()f v v

−

is a relaxation

term that describes the tendency of vehicles to adjust their speed to the given speed set-point

max

(0, )vv



. The functions

()



()g



()f v v

−

()P



are determined by the properties of

the cruise controller of the vehicles (see [16]).

Remarks: (i) Model (2.1), (2.2) does not include non-local terms and has certain characteristics from

the kinematic theory of fluids. Traffic flow is isotropic, as in fluid flow, since the vehicles are

autonomous and do not react to downstream vehicles only (as in conventional traffic), but also to

upstream vehicles.

(ii) There are infinite equilibrium points, namely the points where

()vv







and

()

  



for all





(iii) The selection of

()



()g



has several implications for the characteristics of the traffic flow.

The dynamic viscosity

()



makes the “traffic fluid” act as a Newtonian fluid. However, in contrast

to actual fluids, the dynamic viscosity also satisfies (2.5). For isentropic (or barotropic) flow of gases,

the dynamic viscosity and the pressure are always increasing functions of the fluid density (see the

discussion in [17]). Thus, a traffic fluid can have very different physical properties from those of real

compressible fluids.

Model (2.1), (2.2) is to be studied under the following conditions for all





( , ) = 0

lim

  

→

(2.6)

( , ) =

lim vv





→

(2.7)

( , ) <d

   

+

−

+



. (2.8)

Condition (2.6) expresses the fact that “far downstream and far upstream” the highway is “empty”,

i.e., there are no vehicles. Condition (2.8) expresses the fact that the total mass of the vehicles in the

highway is finite. Let

r

be a given constant length. Using the variable transformation

=rx v





x

and the dimensionless quantities

=rt





=max



−

max





=vv



−





, we obtain the following dimensionless model for

0t

x

( ) = 0



(2.9)

( ) ( ) '( ) = ( ( ) ( ) ) ( )

t x x x x

q w w q w ww P g w w f w

      



+ + −

(2.10)

With constraints



)

( , ) 0,t x R





( , ) ( 1, )w t x b−

for

>0t

x

, where

1R

0b

are

constants,

1()gC

is an increasing function with

( ) > 0gw



for all

( 1, )wb−

:f →

with

(0) = 0f

is a

function with

( ) > 0wf w

for all

0w

, and

222

2 ( 1)

( ) = (1 ) 2( ) (1 )

b b w

q w b b w w

+−

+−+

, for all

( 1, )wb−

. (2.11)

Moreover, the functions



)

: 0,R



→



)

: 0,PR +

→

are



)

( )

10,CR

and satisfy the

following properties:

( ) =

lim



−

→

+

(2.12)

( ) = 0, ( ) = 0P

  

for all

[0,1]





and

( ) 0, ( ) 0P

  



for all

(1, )R





. (2.13)

Model (2.9), (2.10) is studied under the following conditions for all

0t

( , ) = 0

lim

xtx



→

(2.14)

( , ) = 0

lim

xw t x

→

(2.15)

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1ANonlinearHeatEquationArisingfromAutomated-VehicleTrafficFlowModelsDionysiοsTheodosis*,IassonKarafyllis**,GeorgeTitakis*,IoannisPapamichail*andMarkosPapageorgiou*,****DynamicSystemsandSimulationLaboratory,TechnicalUniversityofCrete,Chania,73100,Greece,emails:dtheodosis@dssl.tuc.gr,gtitakis@dssl.tuc...

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