Numerical simulation of air pollution related to traffic flow in urban networks

Transcription

1 Numerical simulation of air pollution related to traffic flow in urban networks L.J. Alvarez-Vázquez a,, N. García-Chan b, A. Martínez a, M.E. Vázquez-Méndez c a Depto. Matemática Aplicada II, Universidade de Vigo. E.I. Telecomunicación Vigo. Spain b Depto. Física, Universidad de Guadalajara. C.U. Ciencias Exactas e Ingenierías Guadalajara. Mexico c Depto. Matemática Aplicada, Universidade de Santiago de Compostela. Escola Politécnica Superior. 272 Lugo. Spain Abstract As it is well known, traffic flow is the main pollution source in many urban areas, where the number of vehicles ranges from many thousands to millions. Thus, estimating the pollution emission rate due to traffic flow in big cities is a very hard task. To approach this environmental issue, in this paper we propose a methodology that consists of combining the 1D Lighthill-Whitham-Richards traffic model for road networks with a classical 2D advection-diffusion-reaction pollution model for the atmosphere. Here, the pollution model uses a source term that takes into account the traffic flow contamination by means of a Radon measure supported on a road network within an urban domain. Furthermore, we establish the existence of solution of the coupled model, and detail a complete numerical algorithm to compute it (mainly, interfacing a Godunov scheme based on the supply-demand method for the traffic model, with a characteristics-lagrange finite element method for the pollution model). Finally, several numerical experiences for a real urban domain (the Guadalajara Metropolitan Area in Mexico) are presented. Corresponding author addresses: lino@dma.uvigo.es (L.J. Alvarez-Vázquez), nestorg.chan@red.cucei.udg.mx (N. García-Chan), aurea@dma.uvigo.es (A. Martínez), miguelernesto.vazquez@usc.es (M.E. Vázquez-Méndez) Preprint submitted to Elsevier May 9, 216

2 Keywords: Traffic flow, Air pollution model, LWR model, Numerical simulation, Supply-demand method 1. Introduction The term air pollutant is used for any substance in the air (solid particles, liquid droplets, or gases) that can cause harmful effects on humans and/or the environment. Air pollutants can be classified into two mean types: primary pollutants (mainly, the carbon monoxide gas from motor vehicle exhaust, and the sulphur dioxide released from factories, but also nitrogen oxides, volatile organic compounds, particulates, toxic metals, ammonia, radioactive pollutants, etc.), and secondary pollutants (ground level ozone is a prominent example, since it is formed in the air when primary pollutants interact). Urban air quality has been listed as one of the world s worst toxic pollution problems in the XXI century, since air pollution is a significant risk factor for a large number of health conditions including respiratory infections, heart disease and cancer. There are many activities or factors which are responsible for releasing pollutants into the atmosphere. These sources can be classified into two major categories: anthropogenic sources (which are mostly related to the burning of multiple types of fuel, such as smoke stacks of power plants, factories and waste incinerators, or mobile sources including motor vehicles, marine vessels, and aircrafts), and natural sources (dust from lands with no vegetation, methane from cattle, smoke from wildfires or emissions from volcanic eruptions). Sources can be also characterized by their shape: area sources (two-dimensional sources of diffuse air pollutant emissions, as can be, for example, the emissions from a forest fire), line sources (one-dimensional sources, for instance, the emissions from the vehicular traffic on a roadway), and point sources (single sources of air pollutant emissions - without geometric dimensions - like the emissions from an industrial plant). In this paper we are interested in the numerical modeling of air pollutant transport from a road network (eventually including other pointwise sources) [2, 24, 16, 25, 7, 15, 26, 9]. In addition to a novel mathematical formulation coupling traffic flow in networks and pollution dispersion models, a full computational algorithm is required to conduct this analysis, because of the complex variables involved, including vehicle emissions, traffic volume, meteorology, and terrain geometry. The basic aim of our proposed model is related 2

3 to calculating air pollutant levels in the vicinity of road networks by considering them as line sources. The model takes into account source characteristics such as traffic volume and vehicle velocities; moreover, roads geometry, surrounding terrain and local meteorology are also addressed. The product of these computations (usually a set of mapped contour lines related to concentrations of air pollutants) will be used by the authors in a forthcoming paper to study techniques of reducing adverse air pollutant concentrations (for example, by redesigning road network geometry, altering speed controls or limiting certain types of vehicles), and to assess in the environmental impact statement involving a major new roadway or land use change which will induce new vehicular traffic. So, related to modeling the pollution due to traffic flow in urban networks, in section 2 we propose a novel well-posed mathematical formulation of the traffic-related air pollution problem coupling a one-dimensional traffic model with a two-dimensional air pollution dispersion model. Section 3 is devoted to demonstrate the existence of solution of this new coupled problem, whose solution can be obtained by the numerical algorithm introduced and detailed in section 4. Final sections of the paper are devoted to present several numerical results and conclusions for a realistic case study posed in the Guadalajara Metropolitan Area (Mexico). 2. Mathematical formulation of the environmental problem We consider an urban domain Ω R 2 including, in addition to N P industrial plants (located at points p k, k = 1,..., N P ), a road network similar to the one illustrated in Figure 1. The network is composed of N R unidirectional roads (segments) that meet at a number N J of junctions, such that the endpoints of each segment either are on the boundary of Ω or touch one of the junctions. Each segment A i Ω, i = 1,..., N R, is modelled by an interval [a i, b i ], and we denote by: σ i : [a i, b i ] A i Ω s σ i (s) = (x i (s), y i (s)) (1) a parametrization of the segment A i preserving the sense of moving on the avenue. 3

4 p 3 p 1 A 4 A 2 A 3 A 1 A 1 A 11 A 12 A 9 A 7 A 5 p 2 A 8 A 6 Figure 1: Schematic diagram of a rectangular domain Ω showing N P = 3 industrial plants, and a network formed by N R = 12 unidirectional roads (including 3 incoming roads, 4 outgoing roads, and N J = 5 junctions) Model of traffic flow of type Lighthill-Whitham-Richards (LWR) In order to model the traffic flow in the road network we consider the conservation law formulation proposed by Lighthill and Whitham [19], and by Richards [23], the so-called LRW model. Then, we denote by ρ i = ρ i (s, t) [, ρ max i ], for (s, t) [a i, b i ] [, T ], the density of cars in the avenue A i, by v i = v i (s, t) the velocity, and by Q i = ρ i v i the corresponding flux. This model is based on assuming that the velocity v i depends exclusively on the density ρ i, and, consequently, there exist a function f i : [, ρ max i ] R, the so-called static relation, such that Q i = f i (ρ i ), for i = 1,..., N R. Thus, the conservation law of the number of vehicles is described by the following system of differential equations: For i = 1,..., N R, ρ i t + f i(ρ i ) s = in (a i, b i ) (, T ), (2) which must be accompanied by the corresponding initial and boundary conditions: 1. Initial conditions: For i = 1,..., N R, we assume known the data functions ρ i (s) verifying: ρ i (., ) = ρ i in [a i, b i ]. (3) 2. Boundary conditions: As already mentioned, the endpoints of each av- 4

5 enue, either are on the boundary of Ω, or coincide with one of the junctions. Let us assume that we know the traffic density at the incoming endpoints of the avenues that are on the boundary. To do this, without loss of generality, we assume that the avenues are numbered so that the incoming ones are the N in first, and that we know functions ρ in i (t), for i = 1,..., N in, such that: ρ i (a i,.) = ρ in i in (, T ). (4a) If at the outgoing endpoints of the avenues, density traffic data are available, then on each outgoing avenue l, with l {1,..., N R } we impose the condition: ρ l (., b l ) = ρ out l in (, T ). (4b) In case of data unavailability, it is assumed that avenues leaving the domain are infinite. Those avenues are modelled by the infinite segment [a l, ) (that is, equations (2) and (3) should be posed, respectively, in (a l, ) (, T ) and [a l, )), although we are only interested in the behaviour of the model along the finite interval [a l, b l ]. Finally, in order to avoid the underdetermination of the system, we prescribe the conservation of the number of cars at all of the junctions. So, for the j-th junction (with j {1,..., N J }) we denote by n j the number of incoming roads, and by m j the number of outgoing roads. To be more precise, we assume that the incoming roads are A νj (1),..., A νj (n j ), and that the outgoing ones are A νj (n j +1),..., A νj (n j +m j ). The conservation of cars in junctions can be expressed by the Rankine-Hugoniot relation (see [6, 11, 17] for a more detailed description): For j = 1,..., N J, n j f νj (k)(ρ νj (k)(b νj (k),.)) = k=1 n j +m j l=n j +1 f νj (l)(ρ νj (l)(a νj (l),.)) in (, T ). (4c) However, the initial/boundary value problem (2)-(4) it is not yet univocally determined. To determine it in a unique way, following [6], we also assume the following rules: (R1) We take into account some preferences of drivers, in the sense that the traffic from incoming roads is distributed on outgoing roads according to fixed coefficients. That is, for each j = 1,..., N J, we know the 5

6 distribution matrix A j = (α j lk ) M m j n j (R) such that, for all l = 1,..., m j, and for all k = 1,..., n j : m j < α j lk < 1, l=1 α j lk = 1, (5) where α j lk represents the percentage of drivers that, arriving from the incoming avenue A νj (k), take the outgoing avenue A νj (n j +l) at the j-th junction. (R2) Respecting rule (R1), drivers make their choices so that the traffic flux is maximized. Admitting that the rules (R1) and (R2) are mandatory, and under certain hypotheses on functions f i and matrices A j, it can be proved that the problem (2)-(4) admits solution. We analyze in detail this issue in next section Model of atmospheric pollution Bearing in mind that our final aim is related to atmospheric pollution generated by vehicular traffic, we take as an indicator of pollution the carbon monoxide (CO) concentration - although we could take nitrogen dioxide (N 2 O), ozone (O 3 ), sulphur dioxide (S 2 O) or even particulate matter (PM) instead. So, the concentration of CO in the domain Ω over the time interval (, T ) is given by the following initial/boundary value problem (see, for instance, [12, 25, 26]): φ + u φ (µ φ) + κφ = F in Ω (, T ), t φ(., ) = φ in Ω, µ φ n φ u n = on (6) S, µ φ n = on S+, where u(x, t) is the wind velocity field, µ(x, t) is the CO molecular diffusion coefficient, κ(x, t) is the rate of the (first order) reaction term, F represents the source of contaminant which, as discussed below, will be given by (8), φ is a known function giving the initial concentration of CO, n denotes the unit outward normal vector to the boundary Ω, and S and S + represent, 6

7 respectively, the inflow and outflow parts of the boundary of Ω, that is, S S + = Ω (, T ) with: S = {(x, t) Ω (, T ) such that u n < }, S + = {(x, t) Ω (, T ) such that u n }. In this work we consider two different types of pollution sources: 1. In metropolises, the main source of CO pollution is related to emissions of vehicles. If the avenues in the domain are considered as subsets of Ω with nonzero measure, CO pollution caused by cars can be modelled by a function taking only non-null values on the avenues (see, for example, [25, 26]), but, unfortunately, this model is not compatible with our onedimensional simulation of vehicle emissions. We will assume here that vehicle emissions are proportional to the traffic flow on the roads, which we will know from the solution of model (2)-(4). Implicitly, this means considering avenues as curves in Ω (sets of measure zero) and, consequently, emissions on the avenues should be included in the model (6) through measures on these sets (see [21]). Thus, we propose to model the source of pollution due to vehicular traffic by: N R F 1 = γ i ξ Ai, i=1 where γ i is a contamination rate, and, for each t [, T ], ξ Ai (t) is the Radon measure (an element of the dual of the space of continuous functions) given by: ξ Ai (t) : C(Ω) R v ξ Ai (t), v = Q Ai (., t) v(.) dσ A i where Q Ai (., t) represents the vehicular flux on the avenue A i. For the parametrization given by (1), we have Q Ai (σ i (s), t) = f i (ρ i (s, t)) with ρ i the solution of model (2)-(4). In this case, and assuming the 7

8 functions to be smooth enough, we have that ξ Ai (t), v = bi a i f i (ρ i (s, t)) v(σ i (s)) σ i(s) ds, v C(Ω) (7) where the lengths of the avenues A i R 2 coincide with the lengths of the segments [a i, b i ] R whenever σ i(s) = We also admit that in the domain Ω there exist N P industrial plants, located at points p k Ω, k = 1,..., N P, each of which discharges into the atmosphere a CO flow whose intensity is given by a known function m k (t) L (, T ). This type of pollution sources are usually modelled by the term (see, for instance, [12, 26]): N P F 2 = m k (t) δ(x p k ), k=1 where δ(x p k ) represents the Dirac measure at point p k, that is, δ(x p k ) : C(Ω) R v δ(x p k ), v = v(p k ) Thus, the source of contamination F for the model (6) is finally given by expression: N R N P F = F 1 + F 2 = γ i ξ Ai + m k δ(x p k ). (8) i=1 3. Mathematical analysis of the model 3.1. The traffic model: Problem (2)-(4) The solution of problem (2)-(4) should be defined in terms of weak entropic solutions of (2), which are admissible weak solutions at every junctions (see the details, for example, in [6] or [1]). In particular, we have the following definition [6, Def. 2.2]: Definition 1. Given functions ρ i L (a i, b i ), i = 1,..., N R, and ρ in i L (, T ), i = 1,..., N in, we say that a collection of functions ρ = (ρ 1,..., ρ NR ), with ρ i C(, T ; L 1 loc (a i, b i )), is an admissible solution of problem (2)-(4) if the following properties are satisfied: 8 k=1

9 1. For i = 1,..., N R, ρ i is a weak entropic solution of (2), that is, for every function ϕ : [a i, b i ] [, T ] R smooth with compact support on (a i, b i ) (, T ), verifies T bi a i ( ϕ ρ i t + f i(ρ i ) ϕ ) dx dt =, x and for every k R and every ϕ : [a i, b i ] [, T ] R smooth, positive with compact support on (a i, b i ) (, T ), verifies T bi a i ( ρ i k ϕ t + sgn(ρ i k)(f i (ρ i ) f i (k)) ϕ ) dx dt. x 2. Initial conditions (3) and boundary conditions (4a) are verified (in a weak sense, as given in [2]). 3. For j = 1,..., N J, the collection of functions ρ νj (1),..., ρ νj (n j +m j ) is a weak solution at junction j, that is, the following equality is verified n j +m j l=1 ( T bνj (l) a νj (l) ( ϕ l ρ νj (l) t + f ν j (l)(ρ νj (l)) ϕ ) ) l dx dt =, x for every ϕ l, l = 1,..., n j + m j, smooth having compact support in (a νj (l), b νj (l)] (, T ) for l = 1,..., n j, and in [a νj (l), b νj (l)) (, T ) for l = n j + 1,..., n j + m j, that are also smooth across the junction, i.e., verifying, for k = 1,..., n j, and l = n j + 1,..., n j + m j, ϕ k (b νj (k),.) = ϕ l (a νj (l),.), ϕ k x (b ν j (k),.) = ϕ l x (a ν j (l),.). 4. Finally, if, for i = 1,..., N, ρ i (., t) is a function of bounded variation for every t, then, the following properties are held at each junction j = 1,..., N J : (i) For k = 1,..., n j, m j f νj (k)(ρ νj (k)(b νj (k),.)) = α j lk f ν j (n j +l)(ρ νj (n j +l)(a νj (n j +l),.)). l=1 (ii) The incoming flux n j k=1 f ν j (k)(ρ νj (k)(b νj (k),.)) is maximum, subject to previous property (i). 9

10 Remark 1. In Definition 1 if, for i = 1,..., N, functions ρ i (., t) are of bounded variation, property 3 guarantees the fulfilment of condition (4c) (see [1, Lemma 5.1.9]). Moreover, properties 4 (i) and 4 (ii) represent, respectively, rules (R1) and (R2). In order to obtain existence of solution for problem (2)-(4), additional hypotheses on fluxes f i and matrices A j should be made. First we remark that, from a theoretical viewpoint, we can assume (without loss of generality) that ρ max i = 1, for i = 1,..., N R. In this case, all the fluxes are equal (f 1 =... = f NR = f), and we make the following assumption on the flux f (which is very reasonable in the context of traffic flow, as can be seen, for instance, in [11] and [17]): (H1) The flux f : [, 1] R is a Lipschitz continuous and concave function satisfying: (a) f() = f(1) =. (b) There exists a unique ρ C (, 1) (denoted critical density) such that f is strictly increasing in [, ρ C ) and strictly decreasing in (ρ C, 1]. (The value C = f(ρ C ) is usually known as road capacity.) For matrices A j = (α j lk ), following [6], we also have to make a technical assumption. In order to do it, if {e 1,..., e nj } represents the canonical basis of R n j, we denote, for l = 1,..., n j, H j l = {e l } (the coordinate hyperplane orthogonal to e l ) and, for l = 1,..., m j, H j n j +l = {α j l }, where α j l = (α j l1,..., αj ln j ) R n j. Let I j be the set of indices i = (i 1,..., i l ), 1 l n j 1, such that i 1 < i 2 <... < i l n j + m j, and for every i I j set l H j i = H j i h. h=1 So, denoting 1 j = (1,..., 1) R n j, we make the following assumption on matrices A j : (H2) The matrices A j = (α j lk ) given in (R1), verify that 1j / (H j i ), for every i I j. Remark 2. In the case of a simple junction j with two incoming roads and 1

11 two outgoing roads (n j = m j = 2), matrix A j is given by ( ) A j β1 β = 2. 1 β 1 1 β 2 In this case, assumption (H2) is equivalent to β 1 β 2. Remark 3. From (H2) we immediately derive that m j n j (the number of outgoing roads should be greater that the number of incoming roads). However, usual junctions with n j 2 and m j = 1 (on-ramps) can be also included in this study. For this type of junctions, the condition in (H2) should be changed by the following (see, for instance, [1, Section 5.2.2]): When not all cars can go through the junction, there exists a yielding rule that describes the percentage of cars crossing the junction, which comes from a particular incoming road. Explicitly: (H2*) For n j 2 and m j = 1, let C j be the amount of cars that can enter the outgoing road. If not all cars can go through the junction, we have a fixed vector q j = (q j 1,..., q j n j ), verifying n j < q j k < 1, q j k = 1, in such a way that q j k Cj gives the amount of vehicles that, coming from the k-th incoming road, enter the outgoing road. We have the following result of existence of solution for problem (2)-(4). It was proved initially in [6], assuming junctions with at most two incoming roads and two outgoing roads, but it was later extended to the general case in [11]. Theorem 1. Consider a flux function f satisfying (H1). If assumption (H2) (or (H2*)) is verified, the problem (2)-(4) has an admissible solution in the sense of Definition 1. k= The atmospheric pollution model: Problem (6) From Theorem 1 we have that ρ i C(, T ; L 1 loc (a i, b i )), for i = 1,..., N R. Moreover, functions f i are continuous and, consequently, measures given by (7) are well defined. In this way, taking into account that we have already 11

12 considered m j L (, T ), j = 1,..., N P, the second member F of the partial differential equation in (6), given by (8), is a well-defined Radon measure. Finally, in order to prove existence of solution for problem (6), we also need to assume additional regularity hypotheses on the problem data. So, we suppose that u (L (Ω (, T ))) 2, ν and κ L (Ω (, T )), and φ L 2 (Ω). For problem (6) we recall the definition of weak solution given in [5, Def. 6.1] for a related problem: Definition 2. Given r, p [1, 2) such that > 3, we say that a function φ L r (, T ; W 1,p (Ω)) is a weak solution of problem (6) if the following r p equality is verified, for all η C 1 (Ω [, T ]) such that η(., T ) = : T = Ω ( y η t N T bi i=1 N P T + k=1 a i ) + ηu φ + ν φ η + κφη dx dt f i (ρ i (s, t)) η(σ i (s), t) σ i(s) ds dt m k (t) η(p k, t) dt + Ω φ (x) η(x, ) dx φη u n dγ. S In this case, we also introduce the concept of transposition solution. In order to do that, we consider the functional spaces: Y = L 2 (, T ; H 1 (Ω)) C(Ω [, T ]), Y T = {η Y : η(t ) = }, YT = {η Y T : η t + L η L (Ω (, T )), η L (S + η ), n L n L (S )}, where L is the adjoint operator of Lφ = u φ (µ φ) + κφ, given by: and L η = (µ η + ηu) + κη, η n L Then, we have the following definition: = (µ η + ηu) n. 12 (9)

13 Definition 3. We say that a function φ is a transposition solution of problem (6) if for some r, p [1, 2) with > 3, φ r p Lr (, T ; W 1,p (Ω)), and the following equality is verified, for all η Y T = Ω N ( φ i=1 N P T + k=1 ) t + L η η T bi a i dx dt + T : S + φ η n L dγ + f i (ρ i (s, t)) η(σ i (s), t) σ i(s) ds dt m k (t) η(p k, t) dt + Ω φ (x) η(x, ) dx. S φ η n dγ Now, we can prove the following existence and uniqueness result: (1) Theorem 2. The problem (6) has a weak solution φ, which is the unique transposition solution of (6). Moreover, φ L r (, T ; W 1,p (Ω)), r, p [1, 2) such that 2 r + 2 p > 3. Proof. The existence of solution can be obtained arguing as in [5, Th. 6.3]. The uniqueness of the transposition solution is a direct consequence of the surjectivity of the mapping: {η Y T : η n L = on S +, η n = on S } η t + L η L (Ω (, T )) that can be obtained by adapting classical technical results from Di Benedetto [8, Th. 4] and Ladyzenskaja et al. [18, Ch. III, Th. 7.1]. Finally, the transposition solution φ of problem (6) is also a weak solution, and consequently, φ L r (, T ; W 1,p (Ω)), r, p [1, 2) such that 2 r + 2 p > Numerical resolution 4.1. LWR model We consider first only a single avenue A Ω defined from the interval I = [a, b] R. So, in order to simplify our notation, we suppress the subscript i in the problem (2)-(4), and approximate its solution using a classical Godunov scheme [14]. To do this, we divide the spatial domain I = [a, b] into M cells I k = [s k 1 2, s k+ 1 2 ] of length s >, and denote by s k = (s k s k+ 1 2 )/2 the

14 midpoint of each cell. The time interval [, T ] is also divided into N N subintervals of length t = T/N, and define t n = n t. Then, we consider the control volume V I k [t n, t n+1 ] and, integrating equation (2) in this domain, we obtain that with: ρ n+1 k = ρ n k t ( ) f n f n, (11) k+ s 1 k ρ n k = 1 s sk+ 1 2 ρ(s, t n ) ds, representing the mean density of traffic in s k 1 2 cell I k at time t n, and f n k± 1 2 = 1 t t n+1 t n f(ρ(s k± 1 2, t)) dt, where f n k+ 1 2 represents the outgoing downstream boundary flux from the cell I k to the cell I k+1, and f n k 1 2 denotes the incoming upstream boundary flux to the cell I k from the cell I k 1. Now, assuming that ρ(s, t) is piecewise constant over the spatial grid, we have that ρ n k = ρ(s k, t n ). Then, considering that for k = 1,..., M, and n = 1,..., N, we already know ρ k = ρ (s k ) and ρ n 1 = ρ in (t n ), the expression (11) becomes a classical numerical scheme (the Godunov s first order method) to solve the problem (2)-(4), once we have decided how to compute the numerical fluxes f n, for k = 1,..., M, n =,..., N 1. In this work k+ 1 2 we propose to approximate these fluxes by the well-known supply-demand method, which consists of computing: f n k+ 1 2 = min{dem( ρ n k), supp( ρ n k+1)}, k = 1,..., M, n =,..., N 1, (12) where: supp : [, ρ max ] R is the supply function given by { C if ρ ρc supp(ρ) = f(ρ) if ρ C ρ ρ max (13) 14

15 dem : [, ρ max ] R is the demand function given by { f(ρ) if ρ ρc dem(ρ) = C if ρ C ρ ρ max (14) Remark 4. Expression (12) for k = M requires the knowledge of data ρ n M+1, for n =,..., N 1. If at the outgoing endpoint there exist available data providing traffic density - that is, if we have boundary condition (4b) - then we can take ρ n M+1 = ρout (t n ). If we do not have those data, it is assumed that there is free traffic downstream and, consequently, we can take supp( ρ n M+1 ) = C. The supply-demand method is mainly based on assuming that the flow at the intersection of two cells is the minimum between the flow that upstream cell can supply and the flow that downstream cell demands. The definitions of supply and demand functions are based on the assumption that the upstream cell can accommodate the maximum flow (C) for free traffic (ρ ρ C ) and must restrict to actual traffic if this is congested (ρ C ρ ρ max ), while the downstream cell demands only the actual flow in this cell if there is free traffic and the maximum flow if traffic is congested. A detailed explanation of the method can be found in the recent monograph of Treiber and Kesting [28]. It is worthwhile remarking here that supp(ρ) and dem(ρ) are monotone functions (non-increasing and non-decreasing, respectively). Thus, the flux ˆf( ρ n k, ρn k+1 ) = f n given by (12) is a non-decreasing function with respect to k+ 1 2 the first argument, and non-increasing function with respect to the second one. This fact makes the numerical method given by (11)-(12) to become a monotone first order scheme and, consequently, if a proper CFL condition is satisfied, we have that (see [4, Section 3.3]): ρ n k ρ max, k = 1,..., M, n = 1,..., N, (provided that the initial condition verifies ρ (x) ρ max in [a, b]), leading to the bound preserving property of the numerical solution Road Networks The main advantage of the supply-demand method (12) when defining the flows in Godunov scheme (11) results of its ability to be implemented on road 15

16 networks with unidirectional avenues. So, each avenue A i in the network is divided into a number M i of cells, and the traffic densities at the midpoints of these cells are calculated as described in previous section. The only difference is when that avenue begins or ends at a junction. If the avenue begins at a junction, then the incoming boundary condition at that point is not known, and the density in the first cell of that avenue (ρ n+1 i,1, n =,..., N 1) needs to be calculated. This computation is done using the expression (11) for k = 1, but in this case it is necessary to specify how to calculate the incoming upstream boundary flux of the first cell (f n, n =,..., N 1) - i, 1 2 note that in (12) this flux is not considered. Similarly, if the avenue ends at a junction, we do not know the outgoing boundary condition at that point, and we cannot assume that downstream there is free traffic, as in the case of an infinite avenue (see Remark 4). Then, in this case it is necessary to redefine the outgoing flux downstream boundary of the last cell (f n i,m i, n = + 1 2,..., N 1). In below paragraphs we can find how to define these numerical flows from the Rankine-Hugoniot relation (4c), taking into account the rule (R1) and the hypothesis (H2). So, let us consider the junction j {1,..., N J }, where, in order to simplify the notation, we will assume that the n j incoming avenues are A 1,..., A nj and the m j outgoing ones are A nj +1,..., A nj +m j (that is, we supose that ν j (l) = l, for l = 1,..., n j + m j ). Then, For i = 1,..., n j, if not all vehicles can enter the junction j, then the demand of the cell M i is weighted by the vector q j = [q1, j..., qn j j ] - see Remark 3 - (if all vehicles may enter the junction, we take q j = (1,..., 1)). This implies that the demand of the cell M i is q j i dem i( ρ n i,m i ). Moreover, the downstream boundary flux of that cell is limited by the capacity of the outgoing avenues to receive it, and specifically by the sum of supply of the first cell of all the outgoing avenues given by n j +m j l=n j +1 supp l( ρ n l,1 ). Thus, we have: f n i,m i = min qj i dem i( ρ n i,m i ), n j +m j l=n j +1 supp l ( ρ n l,1), i = 1,..., n j. (15) According to the rule (R1) we have a matrix A j = (α j il ) that indicates the percentage of drivers that, entering the avenue l, take avenue n j +i. Thus, for i = 1,..., m j the demand of the first cell of the avenue A nj +i is 16

17 f n n j +i, 1 2 given by n j l=1 αj il dem l( ρ n ). Furthermore, the upstream boundary l,m l flux in each of these cells is limited by their own supply and, therefore, { nj } = min α j il dem l( ρ n l,m l ), supp + 1 nj +i( ρ n n j +i,1), i = 1,..., m j. (16) 2 l=1 Under these considerations, boundedness is also preserved in road networks. In fact, arguing as in [4], we have that the numerical solution ρ n i,k obtained by previous method verifies that: ρ n i,k ρ max, i = 1,..., N R, k = 1,..., M i, n = 1,..., N Pollution model Once solved the traffic model (2)-(4) (i.e., once computed the density functions ρ i (s, t), i = 1,..., N R ), we can address the pollution problem (6). In order to solve this problem we use an algorithm combining the method of characteristics for the time discretization with a Lagrange P 1 finite element method for the space discretization, which presents good theoretical convergence results (see the details in [1]). The method of characteristics stems from the equality: Dy Dt = y t + u y, where Dy Dt denotes the total derivative of y with respect to t and u, that is, Dy [ ] (x, t) = y(x(x, t; τ), τ) Dt τ τ=t with τ X(x, t; τ) the characteristic line providing the trajectory of the fluid particle that occupied the position x at time t. This characteristic line can be obtained by integrating the following initial value problem: dx = u(x(x, t; τ), τ), dt (17) X(x, t; t) = x. We denote X n (x) = X(x, t n+1 ; t n ) the position at instant t n of the particle that at the instant t n+1 was in x, and we consider the following approxima- 17

18 tion: φ Dφ + u φ = t Dt φn+1 (x) φ n (X n (x)) (18) t This leads us to approach system (6) by the following semi-discretized problem: Given φ, find φ n+1 W 1,p (Ω), for n =,..., N 1, satisfying φ n+1 φ n X n (µ φ n+1 ) + κφ n+1 t N R N P = γ i ξ n+1 A i + m n+1 k δ(x p k ) in Ω, i=1 k=1 (19) µ φn+1 n φn+1 u n+1 n =, on S, µ φn+1 n = on S+, where a superscript n {,..., N} denotes the corresponding function valued at time t n. We consider now a polygonal approximation Ω h of Ω and we choose an admissible triangulation τ h of it, formed by triangles of diameter lower than h and vertices {x l : l = 1,..., N V } such that the vertices on the boundary of Ω h also lie in the boundary of Ω. Bearing in mind that any piecewise polynomial function in W 1,p (Ω h ) is continuous, we approximate W 1,p (Ω h ) by V h = {v h C(Ω h ) : h T P 1, T τ h } and, from the direct variational formulation of (19), we obtain the following fully discretized approximation of the problem (6): { } } } NR N Given the numerical solution {{ ρ n+1 Mi i,k of the LWR model k=1 (2)-(4), and the initial condition φ h V h, find φ n+1 h V h, for n =,..., N 1, satisfying: Ωh φ n+1 h i=1 n= φ n h Xn v h µ φ n+1 h v h + φ n+1 u n+1 n v h t Ω h S N R + κφ n+1 h v h = Ω h i=1 γ i ( s) i M i k=1 N p + m n+1 k v h (P k ), v h V h, k=1 f i ( ρ n+1 i,k ) v h(σ i,k ) σ i,k (2) 18

19 where σ i,k = σ i (s i,k ), and X n h is the approximation of Xn obtained by solving the problem (17) with the backward Euler scheme, that is, X n h (x) = x t u n (x). As it is well known, by introducing the nodal basis of V h, the problem (2) is equivalent to solving N square linear systems of order N V with symmetric matrix (the same matrix for all the N systems), which can be solved by means of any standard algorithm for linear systems. 5. Numerical results 5.1. A case study: The Guadalajara Metropolitan Area (GMA) The GMA is the second largest metropolitan area in Mexico with a population of 4.5 million inhabitants. An estimated amount of about two million vehicles circulate daily through their streets and avenues, which has significantly increased all the levels of air pollution, to the point that, in most days, the currently permitted levels are exceeded. Air pollution in the GMA has been (and remains being) the subject of numerous studies from very different viewpoints (see, for instance, [25, 22, 3] and the references therein). We have chosen this area to implement the methodology described in the previous sections, and below we present several numerical results. As domain Ω we have taken the area described in Figure 2 on which, in this first approach to the problem, we have considered a road network formed by N R = 12 unidirectional avenues, with N J = 6 junctions, representing the main roads of the GMA LWR model To solve the system (2)-(4) on the road network under consideration, first we need to set the static relations (that is, the functions f i (ρ i )), which, verifying the hypothesis (H1), provide the theoretical flow on each avenue. In this paper we have used the well-known Triangular Fundamental Diagram (TFD) that involves taking f i (ρ i ) = V i ρ i Ii if ρ i ρ Ci, ( Ii T gap 1 ρ ) i if ρ i ρ max Ci < ρ i ρ max i, i 19

20 A9 A6 5 A11 A8 A7 3 A1 A12 6 W 4 A5 A4 2 A3 1 A1 A2 Figure 2: Satellite photo of the Guadalajara Metropolitan Area, where we show the domain Ω for the case study and the road network under consideration. where Vi (Km/h) is the desired velocity in the avenue Ai, Tigap (h) is the time gap between two vehicles, Ii is the number of lanes in the avenue, and ρmax and ρci (u/km) are, respectively, the maximum density of vehicles and i the critical (jam) density. Taking into account that fi must be continuous, we have that: ρci = 1 Vi Tigap + 1 ρmax i, Ci = fi (ρci ) = Ii Tigap + Vi ρmax i. In the experiments shown here, we considered that the twelve avenues present a similar behaviour, so we admit the same theoretical flow at all of them (that is, f1 =... = f12 = f ). The values we have taken for the parameters that define the flow were V = 5 km/h, T gap = h, I = 1 and ρmax = 12 u/km, resulting so ρc = 4.26 u/km and C = u/h. In order to show the behaviour of the solution obtained by solving the 2

21 (s)= (s) 12 (u/km) s (Km) Figure 3: Initial condition for the LWR model on A 1 and A 2 (functions ρ 1(s) = ρ 2(s)), corresponding to a travelling wave with maximum density ρ max = 12 u/km model (2)-(4) with the method described in section 4.1, we performed two types of experiments. In the first of these tests, we vary the values of the matrices and vectors A j and q j in the different types of junctions, to observe the behaviour of the solution in these zones. In the second one, we impose different boundary conditions on the outgoing avenues, to see how these conditions affect the final solution. In all these experiments we assume that no new vehicles enter the network (that is, ρ in 1 = ρ in 2 = ), and that initially there are cars only in the incoming avenues, where impose an initial condition of type travelling wave (in our particular case, ρ 1 = ρ 2 are as shown in Figure 3, and, for i = 3,..., 12, ρ i = ). To observe how this wave moves along the road network, and considering that the avenue lengths range between 1.19 Km (for the shortest avenue) and 12 Km (for the longest one), we choose a simulation time of T =.8 h Traffic behaviour for different types of junctions In this set of experiments we choose a junction of each possible type. In particular, we have chosen the junction 1, where the number of incoming avenues is less than the number of outgoing ones (n 1 = 1 < 2 = m 1 ), the junction 2, where the number of incoming avenues is greater (n 2 = 2 > 1 = m 2 ), and the junction 3, where both numbers are equal (n 3 = 2 = m 3 ). We analyze the behaviour of the solutions (traffic density and flux) in each one 21

22 of these different junctions: - Junction j = 1 (with incoming avenues A 1 and A 2, and outgoing avenue A 3 ): We consider A 1 = (1 1) and q 1 = (.3,.7), which means that, when traffic is congested in A 3, only the 3% of vehicles entering A 3 will proceed from avenue A 1, and the remaining 7% from avenue A 2. As expected, this choice results in a greater accumulation of the vehicular density (solid line) and a reduction in flow (dashed line) in avenue A 1 with respect to avenue A 2 (see Fig. 4). 1 (s,.8) (u/km) s (Km) (s,.8) (u/km) Q 1=f( 1(s,.8)) (u/h) s (Km) Q 3=f( 3(s,.8)) (u/h) 2 (s,.8) (u/km) s (Km) Q 2=f( 2(s,.8)) (u/h) Figure 4: Behaviour of the solution at junction 1: Vehicular density (solid line) and flux (dashed line) in incoming avenues A 1 (up-left) and A 2 (down-left), and in outgoing avenue A 3 (right), corresponding to the time when the initial waves are going through the junction (t =.8 h). - Junction j = 2 (with incoming avenue A 3, and outgoing avenues A 4 and A 5 ): We take now ( ).2 A 2 = and q 2 = (1),.8 22

23 implying that, from the total amount of cars entering by avenue A 3, the 2% takes avenue A 4 and the 8% takes A 5. The numerical solution achieved (see Fig. 5) reflects in a clear way this preference of drivers, showing as the avenue A 5 suffers a higher density (and flow) of traffic than the avenue A 4. 3 (s,.16) (u/km) s (Km) (s,.16) (u/km) Q 3=f( 3(s,.16)) (u/h) s (Km) Q 4=f( 4 (s,.16)) (u/h) 5 (s,.16) (u/km) s (Km) Q 5=f( 5(s,.16)) (u/h) Figure 5: Behaviour of the solution at junction 2: Vehicular density (solid line) and flux (dashed line) in incoming avenue A 3 (left), and in outgoing avenues A 4 (up-right) and A 5 (down-right), corresponding to the time when the initial waves are going through the junction (t =.16 h). - Junction j = 3 (with incoming avenues A 5 and A 1, and outgoing avenues A 8 and A 7 ): In this third case, we consider the traffic distribution given by ( ).7.3 A 3 = and q 3 = (1, 1)..3.7 We note again that the solution obtained (see Fig. 6) corresponds to the expected distribution of traffic: The avenue A 5 is providing more cars than avenue A 1 and, as the 7% of vehicles coming by A 5 choose avenue A 8, this avenue supports a higher density and a greater flow 23

24 than the alternative avenue A 7. 5 (s,.36) (u/km) 1 3 s (Km) Q 5=f( 5 (s,.36)) (u/h) 8 (s,.36) (u/km).4.8 s (Km) Q 8=f( 8 (s,.36)) (u/h) 1 (s,.36) (u/km) 1 2 s (Km) Q 1=f( 1(s,.36)) (u/h) 7 (s,.36) (u/km) 1 s (Km) 3 Q 7=f( 7 (s,.36)) (u/h) Figure 6: Behaviour of the solution at junction 3: Vehicular density (solid line) and flux (dashed line) in incoming avenues A 5 (up-left) and A 1 (down-left), and in outgoing avenues A 8 (up-right) and A 7 (down-right), corresponding to the time when the initial waves are going through the junction (t =.36 h) Traffic behaviour for different boundary conditions on the outgoing endpoints To see the effect that cause the different boundary conditions on the outgoing endpoints of the avenues, we considered three different boundary data: ρ out 6 ρ C (congested traffic at the endpoint of avenue A 6 ), and ρ out 9 = ρ max (saturated traffic at the endpoint of A 9 ), while in A 12 we admit free traffic downstream (which in model (2)-(4) would be represented by an avenue of infinite length). The behaviour of the solution in these three avenues in the instant that the initial waves (as given in Fig. 3) reach the endpoints, is shown in Figure 7. So, in avenue A 6 some accumulation of vehicles occurs because the output capability is exceeded; however, vehicles continue to leave the network in a considerable flow. At avenue A 12 the cars leave the network freely, so that no agglomeration is observed at the endpoint. Finally, the endpoint of avenue A 9 appears fully saturated, being appreciable a maximum density of vehicles due to a null flow. 24

25 6 (s,.56) (u/km) 12 (s,.56) (u/km) 4 8 s (Km) s (Km) Q 6=f( 6 (s,.56)) (u/h) Q 12=f( 12 (s,.56)) (u/h) 9 (s,.56) (u/km) s (Km) Q 9=f( 9 (s,.56)) (u/h) Figure 7: Behaviour of the solution at the outgoing avenues: Vehicular density (solid line) and flux (dashed line) in avenues A 6 (up), A 12 (middle) and A 9 (down), corresponding to the time when the initial waves are leaving the road network (t =.56 h) Pollution model This final section is devoted to present some of the results obtained by simulating the pollution caused by traffic flow in the GMA. For the sake of simplicity we consider traffic flow as the only source of pollution (that is, we take N P = ), and we must remark that: - We assume a constant characteristic wind of magnitude u = 1.5 Km/h and blowing SW. - Molecular diffusion µ of a contaminant in gas phase depends on the thermal velocity v g and on the collisions between particles, so that µ = λ v g /2, being λ the distance between particles. For aerosols (which includes our CO case) typical values are in the order of 1 8 Km 2 /h (cf. [27]). In this paper we take µ = Km 2 /h. - Decay or reaction rate κ is given by κ = 1/τ = v dep /H where τ is the permanence time in the atmosphere, H is the height of the air column, 25

26 and v dep is the deposition rate, which again depends on the particle diameter and on the wind velocity. For aerosols, the characteristic value of τ is about one week (cf. [27]), which leads us to consider κ = h 1. - Vehicle emissions (which are accessible by brand and model) need to be standardized for a section of avenue. In this work we consider emissions from standard passenger cars, taking for all avenues the same constant γ i = 1 6 Kg/(u Km). - For a better interpretation of the obtained results, we presume no initial contamination (φ = ), and simulate the pollution generated over a period of T hours. - The model (6) has been solved on a mesh (triangulation) τ h generated with free software Gmsh [13]. This mesh, which can be seen in Figure 8, consists of 898 elements and 491 nodes, and its maximum diameter verifies h = max T τh h T 1. Km. Concerning the data obtained from the traffic model, we performed two different blocks of experiments: a first one for the case of targeted traffic, and a second one for the usual traffic on any given day Pollution behaviour in situations of targeted traffic In these first set of experiments we took as data the results obtained with model (2)-(4) for the scenario described in the previous section: no new cars can access the network, and nonzero initial conditions are given only on the incoming avenues A 1 and A 2, where travelling waves (as given in Fig. 3) move through the network according to the preferences defined by matrices A j and vectors q j. We simulated four different situations, which correspond to four different combinations of these matrices and vectors. Thus, in the first situation we have chosen that the initial waves leave the road network mainly through the avenue A 6, in the second one through A 9, in the third one through A 12, and in the last one the waves move throughout the whole network and spread out through the three avenues. Pollution levels reached in these four situations are illustrated in Figure 9, where contaminant concentrations obtained at the last time of the simulation period (T =.8 h) are compared. 26

27 18 16 wind direction y x Figure 8: Polygonal domain Ω h corresponding to the GMA (as given in Fig. 2), showing the triangulation τ h used in the numerical tests, the road network, and the wind direction Pollution behaviour for standard vehicular traffic In this second set of experiments we simulate the pollution caused by vehicular traffic on any given day, for example, from 7: to 19: (that is, a simulation period of T = 12 hours. To do this, we assume that initially there is no traffic on the network (ρ i =, i = 1,..., 12), but that at the entering endpoints of the avenues A 1 and A 2 we impose a continuous density ρ in 1 (t) = ρ in 2 (t), given by a periodic function (of wave type) having its maximum during peak hours and its minimum in valley hours. This function, which in practice should be experimentally known, is different if the day is a weekday or a holiday. We consider both types of days (characterized by the functions given in Figure 1), and perform both simulations considering free traffic downstream in the three outgoing avenues, and taking matrices A j and vectors q j that favour vehicles to leave the network through avenue A 6. Then, we solve the traffic model (2)-(4) in these two situations (weekday and holiday) and, with the results obtained there, we solve the pollution model (6) for both situations. Pollution levels reached at different times can be seen in Figures 11 and 27

28 Figure 9: Comparison between the pollution levels reached at the end of simulation (φ(x,.8)) in four different situations: Traffic addressed to leave the network mainly through the avenue A 6 (up-left), mainly through the avenue A 9 (up-right), mainly through the avenue A 12 (down-left), and traffic circulating throughout the whole network and spreading out through the three avenues (down-right). 28

29 in in (s)= (s) 9 (u/km) t (h) Figure 1: Boundary condition for the LWR model on A 1 and A 2 (functions ρ in 1 (t) = ρ in 2 (t) for a weekday (solid line) and for holiday (dashed line) 12. In all these times, small differences between a holiday (less pollution) and a working day (more pollution) can be observed. It can be noticed as the diffusive effect causes a spot on the avenues that moves in the direction of the wind, and it can be also also observed as pollution keeps growing with the passage of time (cars leave the network, but pollution remains). Finally, it should be remarked that, over time, around outgoing avenue A 6 an important spot caused by high traffic flow is formed, because of the greater number of cars leaving the road network through this avenue under free traffic conditions (high flow). 6. Conclusions In this paper the authors have proposed a new model to simulate urban air pollution due to the flow of vehicles on a road network. This model consists of a novel combination of the 1D LWR model for traffic flow with a classical 2D advection-diffusion-reaction pollution model, in such a way that it is mathematically well posed, and the existence of a regular enough solution is guaranteed. In addition, a complete numerical algorithm is proposed to compute the solution of the coupled model, which presents good numerical properties: several numerical experiences for different scenarios in the GMA (Mexico) illustrate the efficiency of our approach. This numerical algorithm 29

30 Figure 11: Comparison between pollution levels φ(x, t) for a holiday (left) and a weekday (right) achieved at times t = 3 hours (up) and t = 6 hours (down). 3

31 Figure 12: Comparison between pollution levels φ(x, t) for a holiday (left) and a weekday (right) achieved at times t = 9 hours (up) and t = 12 hours (down). 31

32 is easy to apply in any real-world situation and, therefore, can predict contamination before any alterations made to the road network or any changes imposed on the traffic volume. This is one of the main advantages of the model (compared to other models where the source of contamination should be given as a known datum), and represents a first step in studying the different alternatives that can be addressed to try to reduce pollution in an urban domain. In this sense, since this novel formulation makes explicit the dependence of the air pollution on the characteristics of road traffic, the model results a very useful state system for possible optimal control problems where one looks for the traffic actions that cause better results in terms of air pollution. The study of these optimal control problems is one of the objectives where the future work of the authors is focused. Acknowledgements This work was supported by funding from project MTM P of Ministerio de Economía y Competitividad (Spain) and FEDER. The second author also thanks the support from Sistema Nacional de Investigadores SNI and Programa de Mejoramiento del Profesorado PROMEP/13.5/ 13/6219 (Mexico). References [1] L.J. Alvarez-Vázquez, A. Martínez, C. Rodríguez, M.E. Vázquez- Méndez. Numerical convergence for a sewage disposal problem. Appl. Math. Model., 25 (21) [2] C. Bardos, A.Y. LeRoux, J.C. Nédélec. First-Order quasilinear equations with boubdary conditions. Commun. Partial Diff. Equations, 4 (1979), [3] S.E. Benítez-García, I. Kanda, S. Wakamatsu, Y. Okazaki, M. Kawano. Analysis of criteria air pollutant trends in three Mexican metropolitan areas. Atmosphere, 5 (214), [4] S. Canic, B. Piccioli, J.Qiu, T. Ren. Runge-Kutta Discontinuos Galerkin Method for Traffic Flux Model on Networks. J. Sci. Comput., 63 (215),

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