A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in

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1 A front tracking ethod for a strongly coupled PDE-ODE syste with oving density constraints in traffic flow Maria Laura Delle Monache, Paola Goatin To cite this version: Maria Laura Delle Monache, Paola Goatin. A front tracking ethod for a strongly coupled PDE-ODE syste with oving density constraints in traffic flow. Discrete and Continuous Dynaical Systes - Series S, Aerican Institute of Matheatical Sciences, 24, 7 (3), pp <hal-933> HAL Id: hal Subitted on 24 Jan 24 HAL is a ulti-disciplinary open access archive for the deposit and disseination of scientific research docuents, whether they are published or not. The docuents ay coe fro teaching and research institutions in France or abroad, or fro public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de docuents scientifiques de niveau recherche, publiés ou non, éanant des établisseents d enseigneent et de recherche français ou étrangers, des laboratoires publics ou privés.

2 A front tracking ethod for a strongly coupled PDE-ODE syste with oving density constraints in traffic flow Maria Laura Delle Monache Paola Goatin Abstract In this paper we introduce a nuerical ethod for tracking a bus trajectory on a road network. The atheatical odel taken into consideration is a strongly coupled PDE-ODE syste: the PDE is a scalar hyperbolic conservation law describing the traffic flow while the ODE, that describes the bus trajectory, needs to be intended in a Carathéodory sense. The oving constraint is given by an inequality on the flux which accounts for the bottleneck created by the bus on the road. The finite volue algorith uses a locally non-unifor oving esh which tracks the bus position. Soe nuerical tests are shown to describe the behavior of the solution. Introduction The first acroscopic odel for traffic flow dates back to the 95s when Lighthill and Whitha [9] and, independently, Richards [2], proposed a fluid dynaics odel to describe traffic flow on an infinite single road, using a non-linear hyperbolic partial differential equation (PDE). This odel is coonly referred to as the LWR odel. The Cauchy proble has then been extended to initial boundary value proble in [2]. More recently several authors proposed odels that track a single vehicle oving in the vehicular traffic. In these odels the single vehicle trajectory is described with an ordinary differential equation(ode), see[4, 8,, 8] and references therein. Indipendently, in the transport engeneering counity, different nuerical ethods that approxiate solutions for probles with oving bottlenecks have been developed, see [2, 3]. In these works the oving constraints are replaced by a sequence of fixed ones and the discontinuity is applied at the upstrea cell interface with respect to the bottleneck position. Moreover, they only deal with with triangular flux diagras. In this article we consider the odel introduced in [5] to odel the effect of urban transport systes, such as buses, in a road network. Fro an analytical point of view, we deal with a hyperbolic conservation law which describes the evolution of the ain traffic (LWR odel), an ODE which describes the bus trajectory and an inequality constraint which odels the bottleneck effect created by the presence of a bus on a road. Existence of solutions to this proble for general BV data was INRIA Sophia Antipolis - Méditerranée, EPI OPALE, 24, route des Lucioles - BP 93, 692 Sophia Antipolis Cedex (France). E-ail: aria-laura.delle onache@inria.fr, paola.goatin@inria.fr.

3 proved in [4]. This odel can be seen as a generalization to oving constraints of the proble consisting in a scalar hyperbolic conservation law with a (fixed in space) flux constraint, introduced and studied in [, 9, ]. The article presents a nuerical ethod for coputing the solutions of this strongly coupled PDE-ODE systes. The results are obtained by cobining a tracking algorith in Lagrangian coordinates which uses a locally nonunifor esh as in [2] and a tracking algorith which reconstructs the bus position through its interaction with the density waves as in [8]. The article is organized as follows. Section 2 contains soe preliinary notations and definitions. Soe theoretical background and the existence theore for Cauchy probles are introduced. In Section 3 we present a finite volue schee with oving esh and the tracking algorith for the solution of the ODE. In Section 4 we present soe nuerical tests which show the effectiveness of our approxiation. 2 Matheatical Model We consider a slow oving large vehicle (e.g. a bus) that reduces the road capacity and generates a oving bottlenck for the surrounding traffic flow. This can be odeled with a PDE-ODE coupled syste consisting in a scalar conservation law representing the traffic flow with a density constraint and an ODE describing the slower vehicle trajectory: t ρ+ x f(ρ) =, (t,x) R + R, ρ(,x) = ρ (x), x R, ρ(t,y(t)) αρ ax, t R +, () ẏ(t) = ω(ρ(t,y(t)+)), t R +, y() = y. Above ρ = ρ(t,x) [,ρ ax ] is the scalar conserved quantity denoting the ean traffic density and ρ ax is the axial density allowed on the road. The flux function f(ρ) : [,ρ ax ] R + is a strictly concave function such that f() = f(ρ ax ) =. It is given by the following flux-density relation f(ρ) = ρv(ρ), where v is a sooth decreasing function denoting the ean traffic speed that in this article is set to be v(ρ) = V( ρ ρ ax ), with V the axial velocity allowed on the road. y = y(t) represents the slower vehicle position, which oves with a speed ω(ρ(t, y(t)+)). The bus velocity w therefore depends on the downstrea traffic density and is given by ω(ρ) = { Vb if ρ ρ. = ρax ( V b V ), v(ρ) otherwise, (2) that it, the slower vehicle oves with a constant speed V b < V as long as it is not slowed down by the downstrea traffic conditions. If this is the case, it will ove at the sae speed of the ain traffic. The coefficient α ],[ gives the reduction rate of the road capacity due to the presence of the large vehicle. For siplicity, in the following we assue that ρ ax = 2

4 V = so that the odel becoes t ρ+ x (ρ( ρ)) =, (t,x) R + R, ρ(,x) = ρ (x), x R, ρ(t,y(t)) α, t R +, ẏ(t) = ω(ρ(t,y(t)+)), t R +, y() = y. (3) The above odel was introduced in [5], in an engineering fraework, to study the effect of urban transport systes in a road network. Its analytical properties were addressed in [4]. 2. The Rieann Proble with oving density constraint Consider (3) with the particular choice y = and ρ (x) = { ρl if x <, ρ R if x >. (4) We want to define the corresponding Rieann solver with oving density constraint. To this end, we rewrite the equations in the bus reference frae (setting X = x V b t), see Figure, right, and we get t ρ+ X (f(ρ) V { b ρ) =, ρ L if X <, ρ(,x) = ρ R if X >, under the constraint ρ(t,) α. (6) Solving proble(5), (6) is equivalent to solve(5) under the corresponding constraint on the flux. f(ρ(t,)) V b ρ(t,) f α (ρ α ) V b ρ α = Fα. (7) where f α (ρ) = ρ ( ρ α) and ρα = α 2 ( V b). We are now ready to define the Rieann solver for (3), (4) following [5, V] and [4]. Denote by R the standard Rieannsolver, i.e. theap(t,x) R(ρ L,ρ R )(x/t)thatgivestheentropysolution of the conservation equation, and let ˇρ α and ˆρ α with ˇρ α ˆρ α, be the solutions of the equation f(ρ) = f α (ρ α )+V b (ρ ρ α ), see Figure. Definition 2. The constrained Rieann solver R α for (3), (4) is defined as follows.. If f(r(ρ L,ρ R )(V b )) > F α +V b R(ρ L,ρ R )(V b ), then (5) R α (ρ L,ρ R )(x/t) = { R(ρL, ˆρ α )(x/t) if x < V b t, R(ˇρ α,ρ R )(x/t) if x V b t, and y(t) = V b t. 2. If V b R(ρ L,ρ R )(V b ) f(r(ρ L,ρ R )(V b )) F α +V b R(ρ L,ρ R )(V b ), then R α (ρ L,ρ R ) = R(ρ L,ρ R ) and y(t) = V b t. 3

5 f(ρ) f(ρ) V b ρ V b F α ˇρ α ρ α α ˆρ α ρ ˇρ α ρ α ˆρ α ρ V b ρ Figure : Flux function: Fundaental diagra, left. Bus reference frae, right. 3. If f(r(ρ L,ρ R )(V b )) < V b R(ρ L,ρ R )(V b ), then R α (ρ L,ρ R ) = R(ρ L,ρ R ) and y(t) = v(ρ R )t. Note that, when the constraint is enforced (point. in the above definition), a nonclassical shock arises, which satisfies the Rankine-Hugoniot condition but violates the Lax entropy condition. Reark The above definition is well-posed even if the classical Rieann solution R(ρ L,ρ R )(x/t) displays a shock at x = V b t. In fact, due to Rankine-Hugoniot equation, we have f(ρ L ) = f(ρ R )+V b (ρ L ρ R ) and hence f(ρ L ) > f α (ρ α )+V b (ρ L ρ α ) f(ρ R ) > f α (ρ α )+V b (ρ R ρ α ). Reark 2 The density constraint ρ(t, y(t)) α does not appear explicitly in Definition 2., and in the following Definition 2.2. It is handled by the corresponding condition on the flux f(ρ(t,y(t))) ω(ρ(t,y(t)))ρ(t,y(t)) F α. (8) The corresponding density on the reduced roadway at x = y(t) is found taking the solution to the equation ( f(ρ y )+ω(ρ y )(ρ ρ y ) = ρ ρ ), α closer to ρ y. = ρ(t,y(t))), see Figure. 2.2 Cauchy Proble: existence of solutions We briefly recall the known results for this type of proble (for details see [4]). We start giving our definition of solution. 4

6 Definition 2.2 A couple (ρ,y) C ( R + ;L BV(R;[,R]) ) W, (R + ;R) is a solution to (3) if. ρ is a weak solution of the conservation law, i.e., for all ϕ Cc(R 2 ;R) (ρ t ϕ+f(ρ) x ϕ)dx dt+ ρ (x)ϕ(,x) dx = ; R + R R (9a) oreover, ρ satisfies the Kružhkov entropy conditions [7] on (R + R) \ {(t,y(t)): t R + }, i.e., for every k [,] and for all ϕ Cc(R 2 ;R + ) and ϕ(t,y(t)) =, t >, ( ρ k t ϕ+sgn(ρ k)(f(ρ) f(k)) x ϕ)dx dt R + R (9b) + ρ k ϕ(,x) dx ; R 2. y is a Carathéodory solution of the ODE, i.e., for a.e. t R + y(t) = y + t ω(ρ(s, y(s)+)) ds ; (9c) 3. the constraint is satisfied, in the sense that for a.e. t R + li (f(ρ) ω(ρ)ρ)(t,x) F α. x y(t)± (9d) Reark that the above traces exist because ρ(t, ) BV(R) for all t R +. Theore 2.3 Let ρ BV(R;[,R]), then the proble () adits a solution in the sense of Definition 2.2. Thecopleteproofcanbefoundin[4]. Itconsistsinconstructingasequenceofapproxiate solutions via the wave-front tracking ethod, and prove its convergence and then check that the liit functions satisfy conditions (9a)-(9d) of Definition Nuerical schee Our ai is to present a nuerical ethod to copute solutions to strongly coupled constrained PDE-ODE probles. Since the solutions of the Rieann proble are known explicitely, it is natural to develop a Godunov-type ethod. The standard Godunov ethod, in principle, could be applied, however, the results produced are not correct, since it will not reproduce all the characteristics of the solutions and it fails to capture the presence of the non-classical shock. This can be overcae by applying a front tracking-capturing ethod which uses a Lagrangian algorith in which the interface is tracked, such as the one in [2], together with a nuerical ethod that tracks at each tie step the slower vehicle trajectory, as in [8]. 5

7 3. Godunov-type schee for hyperbolic PDEs with constraint First, we briefly recall the classical Godunov schee and then we show how it is odified to fulfill our needs. We use the following notation: x n j are the grid points at tie t n with n N and j Z. A coputational cell is given by [x n,x n ] j 2 j+ 2 where x n are the cell interfaces and x n j± j = xn x n is the cell size at tie 2 j+ 2 j 2 t n. The Godunov schee [6] is based on exact solutions to Rieann probles. The ain idea of this ethod is to approxiate the initial datu by a piecewise constant function, then the corresponding Rieann probles are solved exactly and a global solution is siply obtained by piecing the together. Finally, one takes the ean on the cell and proceeds by iteration. Given U(t,x), the cell average of U in the cell j and at tie t n is defined as U n j = x n j x n j+ 2 x n j 2 U(t n,x)dx. () Then the Godunov schee consists of two ain steps:. Solve the Rieann proble at each cell interface x n j+ 2 (Uj n,un j+ ). with initial data 2. Copute the cell averages at tie t n+ in each coputational cell and obtain U n+ j. We reark that waves in two neighbouring cells do not intersect before t n+ = t n + t n if the following CFL condition holds: t n axλ n j Z j+ 2 2 in j Z xn j () where λ n is the wave speed of the Rieann proble solution at the interface j+ 2 x j+ at tie tn. 2 The classical Godunov schee can be expressed in conservative for as U n+ j = Uj n t ( ) F(Uj n,u n x j+) F(Uj,U n j n ), (2) where F(U,V) = f(r(u,v)()) is the corresponding nuerical flux. Since our ai is to track the trajectory of the bus using a Lagrangian algorith a oving esh has to be used. In particular, we develop an algorith which follows at each tie step the bus trajectory and odifies the esh when f(r(ρ L,ρ R )(V b )) > F α +V b R(ρ L,ρ R )(V b ) (3) and the constraint is enforced. This factor needs to be considered when considering a Godunov schee. In particular, if the inequality (3) is satisfied, then the solution oftherieannsolverisnottheclassicaloneandhence, thegodunovscheecannot be applied as it is. We are going to shift grid points locally and, as a consequence, we will have a locally non-unifor esh due to a cell interface oving with the bus trajectory. We will use the superscript new to indicate the quantities that are odifiedattiet n withthegrid. Assuethatattiet n, y n isthebuspositionand y n ]x n,x n ], for soe. When (3) holds, the algorith for the adaptive esh reads as follows: 6

8 If x n y n x n + > 2 2, we replace the point xn with x new 2 2 recopute the cell averages in the cell fro the forula = y n and x n U U new n +(x new = 2 x new with x new = x new x n, see Figure x n 2)U n, (4) x n+ 3 2 x n+ = y n+ x n x n U new U n U n + x n 3 2 x new = y n x n x n x n 3 2 x n y n 2 x n + 2 x n Figure 2: Local shifting of a grid point when x n y n h n + > 2 2. If x n y n x n + 2 2, we adjust the location of the point xn such that + 2 x new = y n and then we ove the point x n to x new at the iddle distance between x n and x new, see Figure 3. We then copute the new cell averages in the cells and + fro the forulas U new = U new + = (x n 2 x new )U n +(x new x n U n + +(x n + 2 x new + x new x new x n + 2)U n, (5) + 2)U n, (6) with x new = x new x new and x new + = x new x n Each tie the constraint is enforced the bus position follows the non-classical shock trajectory: y n+ = x n+ = x new + V ± ± b t n. The other cell interfaces are kept 2 2 unchanged. An explicit forula for the schee can be derived in the following way. Consider for exaple the finite volue cell C in Figure 4 (abcd). Integrate the conservation law over the finite volue: ( t U + x f(u))dxdt =. C 7

9 x n+ 3 2 x n+ 2 x n+ = y n+ x n U n U new U new + x n 3 2 x new 2 x new = y n x n x n 3 2 x n 2 y n x n + 2 x n Figure 3: Local shifting of a grid point when x n y n h n x n+ 3 2 a x n+ = y n+ x n d f U n U n b c e x n 3 2 x n 2 = y n x n + 2 Figure 4: Non unifor finite volue cells. Fro the Green s theore we have which leads to the following U n+ = xn x n+ U n tn x n+ C f(u)dt Udx =, [( ) f(r(u,u n )) V n b R(U,U n ) n ] f(r(u 2,U n )) n. (7) For seplicity, we introduce F which is given by { F(U,V) if f(r(u,v)(vb )) < F F(U,V) = α +V b R(U,V)(V b ), F α otherwise, (8) so that F corrisponds to the F α coputed in (7) when the constraint is active and to the calssical Godunov flux when the constraint is not enforced. After the esh has been resized and adjusted as described earlier in this section, we update the cell averages for all cells with the following conservative forula: U n+ j = xnew j x n+ j U n j tn x n+ j ( F(U n j,u n j+) F(U n j,u n j )). (9) 8

10 3.2 Nuerical ethod for the ODE WeexposeherehowtosolvenuericallytheODE,keepingtrackofthebusposition. At each tie t n we deterine the position y n of the driver by studying interactions between the bus trajectory and the density waves within a fixed cell. We distinguish the two cases: Inequality (3) is satisfied. Then the bus oves with fixed velocity V b and we update the bus position y n+ = y n +V b t n. Inequality (3) is not satisfied. In this case we ipleent the tracking algorith introduced in [8]. We have to distinguish two situations: one when y n [x n,x n j j [ and one when yn [x n j,xn [. In both cases, we check if the 2 j+ 2 wave starting at the cell interface is a shock or a rarefaction and copute the tie of interaction between the wave and the bus trajectory. In the case of the rarefaction the initial and final tie of interaction is coputed and the position of the bus is updated by solving explicitely an ordinary differential equation. According to the new position of the bus, the cell index is updated. 3.3 Nuerical algorith In this section we describe in detail the algorith which is coposed of the following steps: Algorith Algorith for the front tracking finite volue ethod Input data: Initial and boundary condition for the PDE and the ODE, index cell corresponding to the bus position y n. Copute the densities at tie t n+ fro the density values at tie t n using the Godunov flux F: if f(r(u,u n )(V n b )) > F α +V b R(U,U n )(V n b ) then if y n x n > then x n + 2 x new 2 else x n+ 2 2 = y n, copute the new average for U new and update the esh = x new +V b t n. 2 x new = y n, and place the point x new = cell averages for U new x n 3 2 +x new and U+ new and update the esh x n+ + 2 V b t n. end if end if Copute the densities averages at tie t n+ using forula (9). Copute the bus position: if f(r(u n,u n )(V b )) > F α +V b R(U n,u n )(V b ) then y n+ = V b t n +y n else y n coputed with the tracking algorith in [8] end if. Copute the new = x new

11 4 Nuerical results For illustration, we choose a concave fundaental diagra with the following flux function: f(ρ) = ρ( ρ). In this case the Godunov nuerical flux is given by in(f(u),f(v)) if U V, f(u) if V < U < ρ F(U,V) = cr, f ax if V < ρ cr (2) < U, f(v) if ρ cr < V < U. with ρ cr =.5 the density at which the unique axiun of the flux function is attained, f(ρ cr ) = f ax. In this section we present two nuerical tests perfored with the schee previously described. We deal with a road of length paraetrized by the interval [,]. In both the siulations we fix V b =.3, α =.6.. Case I: We consider the following initial data ρ(,x) =.4, y =.5. (2) The solution is given by two classical shocks separated by a non-classical discontinuity, as illustrated in Figures 5 and Tie= Tie= Tie= Tie=.224 Figure 5: Density evolution at different ties corresponding to initial data (2) and a esh grid of 5 points.

12 Figure 6: Density and bus trajectory in the x t plane corresponding to initial data (2) and a esh grid of 5 points. 2. Case II: We consider the following initial data ρ l (,x) =.8, ρ r (,x) =.53, y =.5. (22) The values of the initial conditions create a rarefaction wave followed by a non-classical and a classical shocks on the density, as illustrated in Figures 8 and 7. Figure 7: Density and bus trajectory in a x t plane corresponding to initial data (22) and a esh grid of 5 points.

13 Tie= Tie= Tie= Tie=.769 Figure 8: Evolution of the density at different ties corresponding to initial data (22) and a esh grid of 5 points. 5 Conclusions This article introduces a nuerical ethod for a strongly coupled PDE-ODE proble with oving density constraint which can odel the presence of a oving bottleck on the roads. The PDE describes the evolution of the ain traffic in tie while the ODE describes the bus trajectory. The theoretical fraework of the odel was set up. The density is coputed using a Godunov-type schee with a locally nonunifor esh. Then the position of the bus is recostructed deterining the effects of the interactions with density waves as in [8]. Soe nuerical tests are presented to show the effectiveness of the schee. 6 Acknowledgent This research was supported by the European Research Council under the European Union s Seventh Fraework Progra (FP/27-23) / ERC Grant Agreeent n

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