Riemann solver for a macroscopic double-lane roundabout model

15th IFAC Symposium Control in Transportation Systems June 6-8, 2018. Savona,on Italy 15th IFAC Symposium on Control in Transportation Systems 15th Symposium Control in Transportation Systems JuneIFAC 6-8, 2018. Savona,on Italy June 6-8, 2018. Savona, Italy Available online at www.sciencedirect.com June 6-8, 2018. Savona, Italy 15th IFAC Symposium on Control in Transportation Systems June 6-8, 2018. Savona, Italy

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IFAC PapersOnLine 51-9 (2018) 55–60 Riemann solver for a macroscopic Riemann solver for a macroscopic Riemann solver for a Riemann solver for a macroscopic macroscopic double-lane roundabout model double-lane roundabout model Riemann solver for a macroscopic double-lane roundabout model double-lane roundabout model ∗∗ ∗ Maria Laura Delle Monache Hammond ∗∗ double-lane roundabout ∗ Samuel model Maria Laura Delle Monache Samuel Hammond ∗∗ ∗ ∗∗∗

Maria Monache Hammond Benedetto Piccoli ∗ Samuel ∗∗∗ Maria Laura Laura Delle Delle Monache Samuel Hammond ∗∗ Benedetto Piccoli ∗∗∗ Benedetto Piccoli ∗∗∗ ∗ Benedetto Piccoli Maria Laura Delle Monache Samuel Hammond ∗∗ ∗ Inria, CNRS, Grenoble INP, GIPSA-Lab, ∗∗∗ ∗ Univ. Grenoble Alpes, Benedetto Piccoli Inria, CNRS, Grenoble INP, GIPSA-Lab, ∗ Univ. Grenoble Alpes, Alpes, Inria, CNRS, Grenoble INP, 38000Grenoble Grenoble, France (e-mail: [email protected]). ∗ Univ. Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, GIPSA-Lab, GIPSA-Lab, Grenoble, France (e-mail: [email protected]). ∗∗38000 France (e-mail: [email protected]). RowanGrenoble, University, NJ, USA (e-mail: [email protected]) ∗∗∗38000 38000 Grenoble, France (e-mail: [email protected]). Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, GIPSA-Lab, Rowan University, NJ, USA (e-mail: [email protected]) ∗∗∗ ∗∗ Rowan University, NJ, USA (e-mail: [email protected]) Department of Mathematics, University of Rutgers - Camden, NJ, ∗∗∗ ∗∗ RowanGrenoble, University, NJ, USA (e-mail: [email protected]) 38000 France (e-mail: [email protected]). of Mathematics, University of Rutgers Camden, NJ, ∗∗∗ Department Department of University of USA (e-mail: [email protected]) ∗∗∗ ∗∗ Department of Mathematics, Mathematics, University of Rutgers Rutgers -- Camden, Camden, NJ, NJ, Rowan University, NJ, USA (e-mail: [email protected]) USA (e-mail: [email protected]) USA (e-mail: [email protected]) ∗∗∗ USA (e-mail: [email protected]) Department of Mathematics, University of Rutgers - Camden, NJ, (e-mail: [email protected]) Abstract: In this article, USA we introduce Riemann solver for traffic flow on a roundabout with Abstract: In In this this article, article, we we introduce introduce aa Riemann Riemann solver solver for for traffic traffic flow flow on on aa roundabout roundabout with with Abstract: two lanes. The roundabout is modeled as a sequence of 2for × 1, 1 × 2flow andon2 × 2 junctions. with The Abstract: In this article, weis introduce a Riemann solver traffic a roundabout two lanes. The roundabout modeled as a sequence of 2 × 1, 1 × 2 and 2 × 2 junctions. The two lanes. The roundabout is modeled as a sequence of 2 × 1, 1 × 2 and 2 × 2 junctions. The Riemann solver provides a solution at junctions by taking into consideration traffic distribution, two lanes.solver The roundabout is introduce modeled as a sequence of 2into × 1, 1 × 2flow andon2 × 2 junctions. The Abstract: In this article,aa we a Riemann solver for traffic a roundabout with Riemann provides solution at junctions junctions by taking taking consideration traffic distribution, Riemann solver provides solution at by into consideration traffic distribution, priorities,solver and the maximization ofatthrough flux. We prove existence and traffic uniqueness of the Riemann provides a solution junctions by taking distribution, two lanes. and The the roundabout is modeled as a sequence 2into × 1,consideration 1 × 2 and 2 ×uniqueness 2 junctions. priorities, the maximization of through through flux. Weofprove prove existence and uniqueness of The the priorities, maximization of flux. We existence and of the solution of and the Riemann problem and show some results numerically. priorities, and the maximization of through flux. We prove existence and uniqueness of the Riemann solver provides a solution at junctions by taking into consideration traffic distribution, solution of the Riemann problem and show some results numerically. solution of the Riemann problem and show results numerically. solution of and the Riemann problem and show some some results numerically. priorities, the maximization ofofthrough flux. We prove existence of the © 2018, IFAC (International Federation Automatic Control) Hosting by Elsevierand Ltd.uniqueness All rights reserved. Keywords: Traffic modeling, Roundabouts, Macroscopic models solution of the Riemann problem and show some results numerically. Keywords: Traffic modeling, modeling, Roundabouts, Macroscopic Macroscopic models Keywords: Keywords: Traffic Traffic modeling, Roundabouts, Roundabouts, Macroscopic models models 1. INTRODUCTION Riemann for the different types of junctions. Keywords: Traffic modeling, Roundabouts, Macroscopic models Solver 1. INTRODUCTION INTRODUCTION Riemann Solver for for the the different types types of of junctions. junctions. 1. Riemann Solver We also prove in this the existence unique1. INTRODUCTION Riemann Solver for section the different different types ofand junctions. We also prove in this section the existence and uniqueMacroscopic traffic models were introduced during the We also prove in this section the existence and uniqueness theorem. We also prove in this section the existence and uniqueMacroscopic traffic models were introduced during the 1. INTRODUCTION Riemann Solver for the different types of junctions. ness theorem. Macroscopic traffic were introduced the fifties by Lighthill, Whitham (Lighthill andduring Whitham theorem. • ness In Section 4 we describe the numerical scheme used Macroscopic traffic models models were introduced during the ness theorem. fifties by Lighthill, Whitham (Lighthill and Whitham We also prove in this section the existence and uniqueIn Section Section 44 we we describe describe the the numerical numerical scheme scheme used fifties by Lighthill, Whitham (Lighthill and Whitham (1955)) independently Richards (Richards (1956)). ••• In to find the 4numerical solution of the problem onused the fifties byand Lighthill, Whitham (Lighthill andduring Whitham In Section we describe the numerical scheme Macroscopic traffic models were introduced the (1955)) and independently Richards (Richards (1956)). ness theorem. to find the numerical solution of the problem onused the (1955)) and independently Richards (Richards (1956)). They were the first to propose a hydrodynamics model to find the numerical solution of the problem on the roundabout. (1955)) and independently Richards (Richards (1956)). to find the numerical solution of the problem on the fifties by Lighthill, Whitham (Lighthill and Whitham They were the first to propose a hydrodynamics model • In Section 4 we describe the numerical scheme used roundabout. They were the first propose aascalar hydrodynamics model for traffic flow using hyperbolic Partial • roundabout. In Section 5 we show the results obtained using the They were the first aato tonon-linear propose hydrodynamics model roundabout. (1955)) and independently Richards (Richards for traffic traffic flow using non-linear scalar hyperbolic(1956)). Partial to the 55numerical of the problem on the • In In find Section we show solution the results obtained using for flow using a non-linear scalar hyperbolic Partial Differential Equation (PDE). The PDE equipped with an • Section the obtained numerical introduced in Section 4. using for traffic using non-linear hyperbolic Partial • roundabout. In Section scheme 5 we we show show the results results obtained using the the They wereflow the first ato propose hydrodynamics model Differential Equation (PDE). Theascalar PDE equipped with with an numerical scheme introduced in Section 4. Differential Equation (PDE). The PDE equipped an initial data Equation is commonly referred to as equipped the LWRwith model. numerical scheme introduced in Section 4. Differential (PDE). Thescalar PDE an numerical scheme introduced in Section 4. for traffic flow using a non-linear hyperbolic Partial initial data is commonly referred to as the LWR model. • In Section 5 we show the results obtained using the initial data is commonly as the LWR model. This model was later on referred extendedto networks. initial data Equation is commonly referred toto aswork the on LWR model. 2. scheme MATHEMATICAL MODEL Differential (PDE). The PDE equipped with an This model was later on extended to work on networks. numerical introduced in Section 4. This model extended to work on networks. 2. MATHEMATICAL MODEL MODEL In fact, overwas the later years,on several authors proposed models 2. This model was later extended on networks. initial data commonly referred totoaswork the LWR model. In fact, fact, overis the years,onseveral several authors proposed models 2. MATHEMATICAL MATHEMATICAL MODEL In over the years, authors proposed models on networks that are able to describe the dynamics at In overwas the later years, several authors proposed models This model extended to work networks. on fact, networks that are on able to describe describe the on dynamics at In this article, describe the macroscopic for an 2. we MATHEMATICAL MODELmodel on networks that able to the dynamics at intersections, see forare example et al. Holden In this this article, article, we describe the the macroscopic macroscopic model for for an an on networks that are able toCoclite describe the(2005); dynamics at In we describe model In fact, over the years, several authors proposed models intersections, see for example Coclite et al. (2005); Holden urban double-lane roundabout with four approaches In this double-lane article, we describe the with macroscopic model forand an intersections, see for example et al. (2005); Holden and Risebro (1995); GaravelloCoclite et al. (2016) and reference urban roundabout four approaches approaches and intersections, see for example Coclite et al. (2005); Holden urban double-lane roundabout with four and on networks that are able to describe the dynamics at and Risebro Risebro (1995); (1995); Garavello Garavello et et al. al. (2016) (2016) and and reference reference exits aligned at 90 degrees, see Figure 1. approaches and urban double-lane roundabout with four and In this article, we describe the macroscopic model for an therein. Each of these models consider different types of exits aligned at 90 degrees, see Figure 1. and Risebro Garavello et al. (2016) and reference aligned at see 1. intersections, see for example et al. (2005); Holden therein. Each(1995); of these these modelsCoclite consider different types of exits exits aligned at 90 90 degrees, degrees, see Figure Figure 1. approaches and therein. Each of models consider different types of urban double-lane roundabout with four solutions for different types of junctions, according to the therein. Each of these models consider different types of and Risebro Garavello et al. (2016) and reference solutions for (1995); different types of junctions, according to the solutions for types of according to exits aligned at 90 degrees, see Figure 1. different number of lanes, incoming and outgoing links. solutions for different different types of junctions, junctions, according to the the therein. Each of these models consider different types of different number of lanes, incoming and outgoing links. different number of lanes, incoming and outgoing links. In this article, we focus on a Riemann problem for rounddifferent number lanes, and outgoing solutions for different types junctions, according to the In this this article, weoffocus onincoming Riemann problem forlinks. roundIn we on aaaofRiemann for roundabouts. In particular, we analyze an problem urban double-lane In this article, article, weoffocus focus onincoming Riemann problem forlinks. rounddifferent number lanes, and outgoing abouts. In particular, we analyze an urban double-lane abouts. In we analyzeand an urban double-lane roundabout with four approaches exits at 90 abouts. In particular, particular, we an problem urbanaligned double-lane In this article, we four focusapproaches on analyze a Riemann for roundroundabout with four approaches and exits aligned at 90 roundabout with and exits aligned at degrees. This roundabout can be seen as concatenation of roundabout with four approaches and exits aligned at 90 90 abouts. In particular, we analyze an urban double-lane degrees. This roundabout can be seen as concatenation of degrees. This roundabout can be seen as concatenation of 2degrees. × 1 (”merging”), 1 × 2 (”diverging”) and 2 × 2 (”crossThiswith roundabout can be seen as concatenation of roundabout four approaches and exits aligned at 90 2 × 1 (”merging”), 1 × 2 (”diverging”) and 2 × 2 (”cross2ing”) × × (”diverging”) and 22 × junctions, but11 the can be generalized to a 2ing”) × 11 (”merging”), (”merging”), × 22approach (”diverging”) and × 22 (”cross(”crossdegrees. This roundabout can be seen concatenation junctions, but the the approach can as be generalized toofa ing”) junctions, but approach can be generalized to more general network. The crossing junctions are so called ing”) junctions, but1 the approach can be generalized to aa 2 × 1 (”merging”), × 2 (”diverging”) and 2 × 2 (”crossmore general network. The crossing junctions are so called more general network. The crossing junctions are because the fluxes do not have common destinations. The more general network. The crossing junctions are so so called called ing”) junctions, but approach can bedestinations. generalized to a because the fluxes fluxes dothe not have common destinations. The because the do not have common The main lanes density evolution, as well as the entries and because the density fluxes doevolution, not have common destinations. The more general network. The crossing junctions are so called main lanes as well as the entries and main lanes density evolution, as well as the entries and the exits ones are described by a scalar hyperbolic conmain lanes density evolution, asdestinations. the entries conand because the fluxes not haveby common The the exits exits ones are do described byas awell scalar hyperbolic the ones are described a scalar hyperbolic conservation law. At junction, Riemann problem is the exits areeach described byasathe scalar conmain lanesones density evolution, well as hyperbolic the entries and servation law. At each junction, the Riemann problem is servation law. At each junction, the Riemann problem is uniquely solved using right-of-way and traffic distribution servation law. At each junction, the Riemann problem is the exits ones are described by a scalar hyperbolic conuniquely solved using right-of-way and traffic distribution uniquely solved using right-of-way and traffic parameters. uniquely solved using and traffic distribution distribution servation law. At eachright-of-way junction, the Riemann problem is parameters. parameters. The article is organized as follows: parameters. uniquely solved using right-of-way and traffic distribution The article is organized as follows: The article is as The is organized organized as follows: follows: parameters. • article In Section 2 we introduce formally the mathematical • In Section 2 we introduce formally the the mathematical mathematical •• article In Section 2 we introduce formally The is organized as follows: model by describing accurately thethe network and the In Section 2 we introduce formally mathematical model by describing accurately the network and the model by describing accurately network and mathematical description of thethe traffic evolution on model by describing accurately the network and the the • mathematical In Section 2 wedescription introduce formally the mathematical mathematical description of the traffic evolution on of the traffic evolution on each link and on each junction. mathematical description of the traffic evolution on model by describing accurately the network and the each link and on each junction. link each • each In Section 3 weon introduce the Riemann Solver at junceach link and and each junction. junction. mathematical description of Riemann the trafficSolver evolution on In Section Section 3 we weon introduce the Riemann Solver at junc••• In 3 introduce the at tions. We first introduce some necessary In Section 3first weon introduce the Riemann Solvernotations at juncjunceach link and each junction. tions. We introduce some necessary notations 1. Roundabout modeled in the article. tions. first introduce some notations and weWe describe step by step thenecessary construction of the Fig. Fig. 1. 1. Roundabout Roundabout modeled modeled in in the the article. article. tions. introduce some • and In 3first we introduce the Riemann Solvernotations atof andSection weWe describe step by by step step thenecessary construction ofjuncthe Fig. we describe step the construction the and weWe describe step by step thenecessary construction of the Fig. 1. Roundabout modeled in the article. tions. first introduce some notations Copyright © 2018 IFAC step by step the construction of the 55 Fig. 1. Roundabout modeled in the article. and © we2018, describe 2405-8963 IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2018 IFAC 55 Copyright 2018 responsibility IFAC 55 Control. Peer review© of International Federation of Automatic Copyright ©under 2018 IFAC 55 10.1016/j.ifacol.2018.07.010 Copyright © 2018 IFAC 55

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Definition 2.1. Consider a roundabout with 52 links ii = [ai , bi ] ⊂ R, ai ≤ bi for i = 1, . . . , 52, with b52 = a45 and b44 = a29 , 16 entrance links and 12 exit links. A col52   C0 (R+ ; L1 BV(Ii ) lection of functions (ρi )i=1,...,52 ∈

A roundabout can be seen as a sequence of junctions, and it can be represented by a graph in which roads are described by arcs and junctions by vertices, see Figure 2. The roundabout that we study in the following of this article is equipped with 52 links of which 16 belong to entrances, 12 belong to exits, 16 belong the the outer lane of the circle and 8 belong to the inner lane of the circle. The generalization of the study to an arbitrary number of roads being straightforward. Each link forming the roundabout is modeled by an interval Ii = [ai , bi ] ⊂ R, i = 1, . . . , 52, ai < bi , with b44 = a29 and b52 = a45 to represent the roundabout circles. The evolution of traffic flow along the circle lanes and on

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for every k ∈ R and for all ϕi ∈ C1c (R×Ii ), ϕi ≥ 0, i = 1, . . . , 52 (3) At each junction Ji for i = 1, . . . , 52,   f (ρinc (t, 0−)) = f (ρout (t, 0+)))

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46 30

Ii

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4 45

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where the subscripts inc , out indicates all the links belonging to a junction. In particular, inc indicates the incoming links and out indicates the outgoing ones.

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5

3. RIEMANN PROBLEM AT THE JUNCTION

Fig. 2. Schematics of the roundabout representing the different links and the different junction types.

In this section we describe the construction of the Riemann solver at a junction. Let us first set some notations. We define with ρcr the critical density i.e., the density at which the flux function (2) attains its maximum fmax . Moreover, we introduce the subscripts inc to indicate quantities that belongs to the incoming links on a junction, and out for the outgoing ones Definition 3.1. Let us define the following quantities

the entering and exiting links is described as follows ∂t ρi + ∂x f (ρi ) = 0 (x, t) ∈ R+ × Ii i = 1, . . . , 52, (1) ] is the mean traffic density, where ρi = ρi (t, x) ∈ [0, ρmax i ρmax is the maximal density on each single road and the i flux function is given by the Greenshield fundamental diagram described by the following equation:  ρi  (2) f (ρi ) = ρi vmax,i 1 − max ρi with vmax,i the maximal speed on each link. In the roundabout that we are modeling, there are 3 types of junctions: merge junction (2 incoming and 1 outgoing roads), diverge junction (1 incoming and 2 outgoing links) and crossing junctions (2 incoming and 2 outgoing), see Figure 2 for the different locations of the junctions and Figure 3 for a more detailed representation of the different types of junctions used in this study. 1

2 3

2 Merge junction

1

(1) For every l ∈ {inc} define  if 0 ≤ ρl ≤ ρcr,l f (ρl ) max γinc (ρl ) = fmax,l if ρcr,l ≤ ρl ≤ ρmax,l ;

(2) for every j ∈ {out} define  fmax,j if 0 ≤ ρj ≤ ρcr,j max γout (ρj ) = f (ρj ) if ρcr,j ≤ ρj ≤ ρmax,j .

3

2

(5)

(6)

Moreover, let us fix a matrix A belonging to the set of matrices:    A := A = a1,j j ∈ {out} : 0 ≤ a1,j ≤ 1, a1,j = 1

3

j∈{out}

1

Diverge junction

Ii

for every ϕi ∈ C1c (R+ × Ii ), i = 1, . . . , 52 such that ϕi ≥ 0. (2) ρi satisfies the Kruˇzhkov entropy condition (Kruzhkov (1970)) on (R × Ii ), that is,    |ρi − k|∂t ϕi + sgn(ρi − k)· R+ I i  (f (ρi ) − f (k))∂x ϕi dxdt (4)  + |ρi,0 − k|ϕi (0, x)dx ≥ 0

11

51

44

R+

39 25

50

26

(1) ρi is a weak solution on Ii , i.e., ρi : [0, +∞[×Ii → [0, ρmax ], such that i     ρi ∂t ϕi + f (ρi )∂x ϕi dxdt = 0 (3)

Exits Entries External circle Internal circle 1 × 2 junctions 2 × 1 junctions 2 × 2 junctions

13

15

i=1

is an admissible solution to (1) equipped with initial data ρi (0, x) = ρi,0 (x) if

(7) and a priority vector p = (p1 , p2 ) ∈ R2 with pl > 0, 2  pl = 1, indicating priorities among incoming roads.

4

Crossing junction

l=1

Fig. 3. Different types of junctions modeled

Moreover, we define a function τ as follows. For details, see Garavello et al. (2016). 56

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Definition 3.2. Let τ : [0, ρmax ] → [0, ρmax ] be the map such that

Remark 1. In Sections 3.1 and 3.2 priorities were assigned to links 1 and 2 respectively. We note that the solutions of the Riemann Solver are symmetric with respect to the priority. Therefore the results would have been the same if the other links would be given priority.

• f (τ (ρ)) = f (ρ) for every ρ ∈ [0, ρmax ] • τ (ρ) = ρ for every ρ ∈ [0, ρmax ] \ {ρcr } We are now ready to describe the construction of the Riemann Solver for different types of junctions. For clarity purposes in this section we will describe the solution of the Riemann Solver in terms of fluxes. The equivalent density solution is described in Theorem 1. Fix ρ1,0 , . . . , ρ52,0 ∈ [0, ρmax ]. Consider a Riemann probi lem at a junction Ji  ∂t ρi + ∂x f (ρi ) = 0 (8) ρi (0, ·) = ρi,0 i ∈ 1, . . . , 52

3.3 Crossing junction Let us consider finally a crossing junction , i.e. a junction with two incoming and two outgoing links, see Figure 3, right. In this particular setting the flux from link 1 is allowed only on link 4 and the flux from link 2 is allowed only on link 3. Let us fix constants ρ1,0 , ρ2,0 , ρ3,0 , ρ4,0 ∈ [0, ρmax ] for i = 1, 2, 3, 4, the maximal capacity of the i junction Γmax and a priority parameter p. The Riemann solver RS(ρ1,0 , ρ2,0 , ρ3,0 , ρ4,0 ) = (ρˆ1 , ρˆ2 , ρˆ3 , ρˆ4 ) at the junction is constructed in the following way.

A solution to the Riemann problem at Ji is defined as follows.

(1) Compute:

3.1 Merge junction

max γ1max = γinc (ρ1,0 ),

Let us consider first a merging junction, i.e. a junction with two incoming and one outgoing road, see Figure 3, left. Let us fix constants ρ1,0 , ρ2,0 , ρ3,0 ∈ [0, ρmax ] for i i = 1, 2, 3, and a priority parameter p. The Riemann solver RS(ρ1,0 , ρ2,0 , ρ3,0 ) = (ρˆ1 , ρˆ2 , ρˆ3 ) at the junction is constructed in the following way.

max γ2max = γinc (ρ2,0 ), max γ3max = γout (ρ3,0 ) max γ4max = γout (ρ4,0 ) (2) Set

γ1,4 = min(γ1max , γ4max ) and

(1) Compute :

γ2,3 = min(γ2max , γ3max )

max γ1max = γinc (ρ1,0 ), max γ2max = γinc (ρ2,0 ),

γ3max

=

(3) Then two situations can occur • If γ1,4 + γ2,3 ≤ Γmax then

max γout (ρ3,0 )

γˆ1 = γˆ4 = γ1,4 and

(2) Fix: γˆ3 =

57

min(γ1max

+

γ2max , γ3max ),

γˆ2 = γˆ3 = γ2,3 • else γˆ1 = γˆ4 = min(γ1,4 , max(Γmax −γ2,3 , pΓmax )) and

γˆ1 = min(γ1max , max(ˆ γ3 − γ2max , pˆ γ3 ))

γˆ2 = γˆ3 − γˆ1

γˆ2 = γˆ3 = Γmax − γˆ1

(3) Set γˆinc = (ˆ γ1 , γˆ2 ) and γˆout = (ˆ γ3 )

(4) Set γˆinc = (ˆ γ1 , γˆ2 ) and γˆout = (ˆ γ3 , γˆ4 )

3.2 Diverge junction

Finally the following result holds. Theorem 1. Consider a junction J and fix a priority parameter p ∈]0, 1[ in case of merge junction, a distribution matrix A = [α, 1 − α] in case of diverge junctions, and the maximal capacity of the junction Γmax and a priority parameter p ∈]0, 1[ in case of crossing junction. Let us define with ρinc (ρout ) all the densities belonging to the incoming (outgoing) links. For every ρinc,0 , ρout,0 , there exists a unique admissible solution (ρinc (t, x), ρout (t, x)) in the sense of Definition 2.1, compatible with the Riemann solver proposed in Section 3. More precisely, there exists a unique n-tuple of functions (ˆ ρinc , ρˆout ) such that RS(ρinc,0 , ρout,0 ) = (ˆ ρinc , ρˆout ):  {ρinc,0 }∪]τ (ρinc,0 ), 1] if 0 ≤ ρinc,0 ≤ ρcr , ρˆinc = [ρcr , 1] if ρcr ≤ ρinc,0 ≤ ρmax ,

We consider a diverging junction, i.e. a junction with one incoming and two outgoing links, see Figure 3, center. Let us fix constants ρ1,0 , ρ2,0 , ρ3,0 ∈ [0, ρmax ] for i = 1, 2, 3, i and a distribution matrix A = [α, 1 − α]. The Riemann solver RS(ρ1,0 , ρ2,0 , ρ3,0 ) = (ρˆ1 , ρˆ2 , ρˆ3 ) at the junction is constructed in the following way. (1) Compute: max γ1max = γinc (ρ1,0 ), max γ2max = γout (ρ2,0 ), max γ3max = γout (ρ3,0 ).

(2) Then γˆ1 = min(γ1max , γ1 γˆ2 = αˆ

γ2max γ3max , ) α 1−α

f (ˆ ρinc ) = γˆinc and (9)  [ρcr , 1] if 0 ≤ ρout,0 ≤ ρcr , ρˆout = {ρout,0 } ∪ [0, τ (ρout,0 )[ if ρcr ≤ ρout,0 ≤ ρmax , f (ˆ ρout ) = γˆinc (10)

γˆ3 = (1 − α)ˆ γ1

(3) Set γˆinc = (ˆ γ1 ) and γˆout = (ˆ γ2 , γˆ3 ) 57

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• ∆t is the time step computed such that ∆t maxi vmax,i ≤ c∆x • (tn , xj ) = (n∆t, k∆x) for n ∈ N and k ∈ Z are the grid points

For the incoming road, the solution is given by the wave (ρinc,0 , ρˆinc ), while for the outgoing road, the solution is given by the wave (ˆ ρout , ρout,0 ). Proof. Notice that for the case of merge and diverge junction the Riemann solver is the same as that proposed in Garavello et al. (2016) in Sections 3.2.2 and 3.2.3. In particular, the Riemann solver gives rise to a unique solution to every Riemann problem, see Section 4.2.2 of Garavello et al. (2016). Consider now a crossing junction. Since the equality γˆ1 = γˆ4 and γˆ2 = γˆ3 must hold for every solution, the problem can be solved focusing only on incoming links 1 and 2. In this setting, after restrictions as in (2) are applied, then the solution is defined as for merge junctions. Then the Riemann solver gives rise to a unique solution.

The scheme used for solving equation (1) is the Godunov scheme as introduced in Godunov (1959) and it is based on exact solutions to the Riemann problem. Under the CFL condition, Courant et al. (1967) it holds:   1   ∆t max λk+ 12  ≤ ∆x (11) k∈Z 2 where λk+ 12 is the speed of the wave of the Riemann problem solution at the interface x1 22 at the time tn . The numerical scheme can be written as  ∆t  n n n n n ρn+1 g(ρ = ρ − , ρ ) − g(ρ , g ) . (12) k k k+1 k−1 k k ∆x The numerical flux g takes in general the following expression   min f (z) if u ≤ v z∈[u,v] (13) g(u, v) =  max f (z) if v ≤ u

To guarantee that a Riemann solver provides a good solution for all times we give the following: Definition 3.3. A Riemann solver RS is consistent if for every ρinc,0 , ρout,0 we have RS(ρinc,0 , ρout,0 ) = RS(RS(ρinc,0 , ρout,0 )).

z∈[v,u]

Boundary conditions by setting:

In particular consistency implies that the solution to the Riemann problem satisfies for a.e. t > 0: (ρinc (t, 0−), ρout (t, 0+)) = RS(ρinc (t, 0−), ρout (t, 0+)). We have the following: Theorem 2. The Riemann solver proposed in Section 3 is consistent.

For the incoming roads we proceed

 ∆t  n n g(ρ1 , ρ1 ) − g(ρn1 , ρn2 ) ∆x while for the outgoing ones, we set  ∆t  n n n n n g(ρ − = ρ ) − g(ρ , ρ ) . , ρ ρn+1 −1 K K K K K Ki i i i i i ∆x ρn+1 = ρn1 − 1

Proof. For merge and diverge junctions, the results are given in Section 4.2.2 of Garavello et al. (2016). For crossing junctions, the solution to the Riemann problem can be interpreted as a merge junction for the incoming roads 1 and 2 as for the proof of Theorem 1. Then consistency condition follows from the same property for merge junction.

Conditions at the junctions Each road is divided in Ki cells numbered from 1 to K. From the incoming roads which are connected at the junction at the right endpoint, we set  ∆t  n ρn+1 γˆinc − g(ρnKi −1 , ρnKi ) Ki = ρKi − ∆x while for the outgoing ones, connected at the junctions at the left point, we set  ∆t  n n g(ρ1 , ρ2 ) − γˆout = ρn1 − ρn+1 1 ∆x where γˆinc and γˆout are the flux computed in Section 3.

4. NUMERICAL SCHEME In this section, we describe the numerical scheme used to solve problem (8).

5. NUMERICAL RESULTS

4.1 Network topology

For illustration, we choose a concave fundamental diagram as introduced in (2) with the following values for the parameters. vmax,i = 1, ρmax = 1, L = 1, ρcr = 0.5, T = 2.5 (14) For the sake of simplicity we show simulations with normalized parameters. The extension with general values is straightforward. We choose the following initial conditions:

The roundabout will be modeled by • eight roads from the inner circle (I45 , . . . , I52 ) • sixteen roads for the outer circle (I29 , . . . , I44 ) • eight exit roads, which results in twelve links (I17 , . . . , I28 ), for details see Figure 2 • eight entrances with sixteen links indicated by (I1 , . . . , I16 ), see Figure 2 Moreover, from the topology it can be deduced that the intersections on the roundabout can be represented by 2 × 1, 1 × 2 and 2 × 2 junctions.

ρi (0, x) = 0.4 · ρmax i

for i = 1, 5, 9, 13

(15) (16)

ρi (0, x) = 0.3 ·

for i ∈ [45, . . . , 52] for i ∈ [29, . . . , 44]

ρi (0, x) = 0.2 ·

4.2 Scheme

ρmax i ρmax i

(17)

The results obtained are showed in Figures 4, 5, 6, 7. As example we show the evolution of the density in an entrance link, an exit link, a link on the inside circle and a link on the outside one. In all of them we can see the evolution of the density during the simulation time.

We define a numerical grid in (0, T )×R using the following notation: • ∆x is the fixed grid space 58

2018 IFAC CTS June 6-8, 2018. Savona, Italy

Maria Laura Delle Monache et al. / IFAC PapersOnLine 51-9 (2018) 55–60

1

1

Time = 0 Time = 0.5 Time = 1.25 Time = 2 Time = 2.5

Density [veh/km]

0.8 0.7

0.9 0.8

Density [veh/km]

0.9

0.6 0.5 0.4 0.3 0.2

0.7 0.6 0.5 0.4

Time = 0 Time = 0.5 Time = 1.25 Time = 2 Time = 2.5

0.3 0.2

0.1

0.1

0

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

Space [km]

0.6

0.7

0.8

0.9

1

1

Time = 0 Time = 0.5 Time = 1.25 Time = 2 Time = 2.5

0.7

Time = 0 Time = 0.5 Time = 1.25 Time = 2 Time = 2.5

0.9 0.8

Density [veh/km]

0.8

Density [veh/km]

0.5

Fig. 8. Evolution of the density on a roundabout entrance.

1 0.9

0.6 0.5 0.4 0.3 0.2

0.7 0.6 0.5 0.4 0.3 0.2

0.1

0.1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

0

0.1

0.2

0.3

Space [km]

0.4

0.5

0.6

0.7

0.8

0.9

1

Space [km]

Fig. 5. Evolution of the density on a roundabout exit.

Fig. 9. Evolution of the density on a roundabout exit.

1

1

Time = 0 Time = 0.5 Time = 1.25 Time = 2 Time = 2.5

0.8 0.7

Time = 0 Time = 0.5 Time = 1.25 Time = 2 Time = 2.5

0.9 0.8

Density [veh/km]

0.9

Density [veh/km]

0.4

Space [km]

Fig. 4. Evolution of the density on a roundabout entrance.

0.6 0.5 0.4 0.3 0.2

0.7 0.6 0.5 0.4 0.3 0.2

0.1

0.1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

0

0.1

0.2

0.3

Space [km]

0.4

0.5

0.6

0.7

0.8

0.9

1

Space [km]

Fig. 6. Evolution of the density on a roundabout link in the inside lane of the circle.

Fig. 10. Evolution of the density on a roundabout link in the inside lane of the circle.

1

1

Time = 0 Time = 0.5 Time = 1.25 Time = 2 Time = 2.5

0.8 0.7

0.9 0.8

Density [veh/km]

0.9

Density [veh/km]

59

0.6 0.5 0.4 0.3 0.2

0.7 0.6 0.5 0.4

Time = 0 Time = 0.5 Time = 1.25 Time = 2 Time = 2.5

0.3 0.2

0.1

0.1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

0

Space [km]

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Space [km]

Fig. 7. Evolution of the density on a roundabout link in the outside lane of the circle.

Fig. 11. Evolution of the density on a roundabout link in the outside lane of the circle.

Shown here is an example where a congestion created is generating a traveling jam. The following initial conditions were chosen:

The results obtained are showed in Figures 8, 9, 10, 11. As example we show the evolution of the density on an entrance link, an exit link, a link on the inside circle and a link on the outside one. In all cases, we can see the evolution of the density during the simulation time. The second example illustrates the creation of congestion. It shows the creation of a shock-wave moving upstream on the roundabout circle.

ρi (0, x) = 0.8 · ρmax i

for i ∈ [1, . . . , 16]

(18)

ρi (0, x) = 0.5 ·

for i ∈ [29, . . . , 44]

(20)

ρi (0, x) = 0.5 ·

ρmax i ρmax i

for i ∈ [45, . . . , 52]

(19)

59

2018 IFAC CTS 60 June 6-8, 2018. Savona, Italy

Maria Laura Delle Monache et al. / IFAC PapersOnLine 51-9 (2018) 55–60

6. CONCLUSION

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This article introduces a model for junctions on a roundabout with double lanes and 4 entrances and 4 exits. The junction flow distribution at the junction is solved by using distribution and priority parameters. We proved existence and uniqueness of solutions for the Riemann problem and we solved the problem numerically using the Godunov scheme. Some numerical tests are presented. ACKNOWLEDGEMENTS This research was supported by the NSF grant CNS #1446715 by KI-Net “Kinetic description of emerging challenges in multiscale problems of natural sciences”. REFERENCES Bretti, G., Natalini, R., and Piccoli, B. (2006). Numerical approximations of a traffic flow model on networks. NHM, 1(1), 57–84. Chitour, Y. and Piccoli, B. (2005). Traffic circles and timing of traffic lights for cars flow. Discrete and Continuous Dynamical Systems Series B, 5(3), 599. Coclite, G.M., Garavello, M., and Piccoli, B. (2005). Traffic flow on a road network. SIAM journal on mathematical analysis, 36(6), 1862–1886. Colombo, R.M., Goatin, P., and Piccoli, B. (2010). Road networks with phase transitions. Journal of Hyperbolic Differential Equations, 7(01), 85–106. Courant, R., Friedrichs, K., and Lewy, H. (1967). On the partial difference equations of mathematical physics. IBM journal of Research and Development, 11(2), 215– 234. Delle Monache, M.L., Goatin, P., and Piccoli, B. (2017). Priority-based riemann solver for traffic flow on networks. Communications in mathematical sciences. To appear. Delle Monache, M.L., Obsu, L.L., Goatin, P., and Kasa, S.M. (2014a). Traffic flow optimization on roundabouts. Procedia-Social and Behavioral Sciences, 111, 127–136. Delle Monache, M.L., Reilly, J., Samaranayake, S., Krichene, W., Goatin, P., and Bayen, A.M. (2014b). A pdeode model for a junction with ramp buffer. SIAM Journal on Applied Mathematics, 74(1), 22–39. Garavello, M., Han, K., and Piccoli, B. (2016). Models for vehicular traffic on networks, volume 9 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO. Garavello, M. and Piccoli, B. (2009). Conservation laws on complex networks. Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 26(5), 1925–1951. Godunov, S. (1959). A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Sbornik: Mathematics, 47(8-9), 357– 393. Greenshields, B.D., Bibbins, J., Channing, W., and Miller, H. (1935). A study of traffic capacity. In Highway research board proceedings, volume 14. Holden, H. and Risebro, N.H. (1995). A mathematical model of traffic flow on a network of unidirectional roads. SIAM Journal on Mathematical Analysis, 26(4), 999–1017. 60