
Vehicles in single-lane road
.Solutions of similarity-type for a nonlinear non-classical Stefan problem with temperature-dependent thermal conductivity and a Robin boundary condition are obtained. The analysis of several particular cases are given when the thermal conductivity L(f) and specific heat N(f) are linear in temperature such that L(f)=α+δf with N(f)=β+γf. Existence of a similarity type solution also obtained for the general problem by proving the lower and upper bounds of the solution.
Citation: Lazhar Bougoffa, Ammar Khanfer. Solutions of a non-classical Stefan problem with nonlinear thermal coefficients and a Robin boundary condition[J]. AIMS Mathematics, 2021, 6(6): 6569-6579. doi: 10.3934/math.2021387
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Solutions of similarity-type for a nonlinear non-classical Stefan problem with temperature-dependent thermal conductivity and a Robin boundary condition are obtained. The analysis of several particular cases are given when the thermal conductivity L(f) and specific heat N(f) are linear in temperature such that L(f)=α+δf with N(f)=β+γf. Existence of a similarity type solution also obtained for the general problem by proving the lower and upper bounds of the solution.
In this paper we deal with microscopic modeling of traffic flow, focusing on lane changing dynamics. In particular we study a second order model for one lane that combines two different interaction terms and we describe the extension to the multi-lane case giving particular attention at the two-lane case.
The interest in the dynamics of traffic flow dates back to the first half of the twentieth century and the related mathematical literature is quite large. An overall view can be found, for instance, in the book by Haberman [10] and in the survey paper by Helbing [11].
There are various points of view for modeling traffic flow. In this paper we concentrate on the microscopic approach that is based on the dynamics of individual vehicles considering the individual behaviour of each driver. A typical microscopic model is the Car Following model or Follow the Leader model (FtL) based on the idea that the dynamics of each vehicle (follower) depends on the vehicle in front (leader) and therefore the other vehicles do not affect it. These models are normally for single-lane roads [4,6,14]. A typical Follow the Leader model can be described as follows. In a single-lane with
{˙xn(t)=vn(t)n=1,…,N˙vn(t)=a(xn(t),xn+1(t),vn(t),vn+1(t))n=1,…,N−1˙vN=w(t) | (1) |
where
Many single-lane car following models have been developed and applied to study traffic dynamics. Here we recall some models that will be useful in the following.
The Follow the Leader model, introduced in [26,27], assumes that each vehicle modifies its velocity based on the distance (headway)
{˙xn(t)=vn˙vn(t)=βnvn+1−vn(xn+1−xn)2. | (2) |
The optimal velocity model (OVM) of Bando et al. [3,2] in which a driver aims to a desired velocity function
{˙xn(t)=vn˙vn(t)=αn(V(xn+1−xn)−vn) | (3) |
with appropriate coefficients
We mention also some interesting works. Pipes proposed [25] a traffic model in which each vehicle maintains a certain prescribed "following distance" from the preceding vehicle; the generalized force model (GFM) by Helbing and Tilch [13] in which the optimal velocity function is obtained calibrating the parameters with the observed data; the full velocity difference model (FVDM) by Jiang et al. [17] that predicts delay time of car motion and kinematic wave speed at jam density; the optimal velocity difference model (OVDM) by Peng et al. [24] where a new term is introduced involving the optimal velocity functions and the vehicles
Another type of microscopic model is given by lane changing models which provide for the possibility of changing lanes according to the analysis of some factors that intervene in the decision process, for example the need, opportunity and safety of a lane change [7,29]. The interest in modeling vehicle lane changing is due to the effects that it induces in traffic flow, for instance in bottleneck discharge rate and in the stop & go oscillations. Here we recall some works. Cassidy and Rudjanakanoknad [5] showed that when traffic density upstream of a busy merge increases beyond a critical value, vehicles manoeuvre toward faster lanes causing traffic breakdown and "capacity drop" of the road; Zheng et al. [30] showed that lane changing are responsible for transforming subtle localized oscillations to substantial disturbances; Klar and Wegener [21,20] developed a model based on reaction thresholds from which they derived a kinetic model; Song and Karni [28] proposed a macroscopic model in which the acceleration terms take lead from microscopic car-following models, and yield a non-linear hyperbolic system with viscous and relaxation terms; Herty et al. [15] proposed a macroscopic model, which accounts for lane-changing on motorway, based on a two-dimensional extension of the Aw and Rascle and Zhang macroscopic model for traffic flow; Gong et al. [9] presented a finite dimensional hybrid system based on the continuous Bando-Follow-the-Leader dynamics coupled with discrete events due to lane changing; Goatin and Rossi [8] developed a macroscopic model for multi-lane road networks with discontinuities both in the speed law and in the number of lanes; Hodas and Jagota presented in [16] a microscopic model for multi-lane dynamics where each car experiences a force resulting from a combination of the desire of the driver to attain a certain velocity and change of the force due to cars interactions; Kesting et al. [19] proposed a general model to derive lane changing rules for discretionary and mandatory lane changes for a wide class of car following models; Lv et al. in [22] extended the continuous single-lane models to simulate the lane changing behaviour on an urban roadway with three lanes and in [23] proposed a model where lane changing is not instantaneous but is a continuing process which can affect the following cars; Zheng et al. in [31] analysed the effects of lane changing in the driver behaviour.
This paper proposes the study of a second order microscopic model combining models 2 and 3 for reproducing traffic flow and its extension to the multi-lane case with simple lane changing conditions in order to study its stability under perturbations. In Section 2 we introduce the model for a single-lane and we study its stability in the linearized case, then we show numerical tests making comparisons with model 3. In section 3 we describe the extension of the model to the two-lane case studying its stability around the equilibrium when a lane is perturbed. We present some numerical tests that confirm the predictions of the linear stability analysis. Finally, in section 4, we illustrate the generalization of the model to the generic multi-lane case.
In this section we describe the main mathematical model we use in this paper. Consider a homogeneous population of
The dynamical equations of the system are obtained combining two interaction terms. The first one is the interaction term related to the model 3 [2,3]. It is a relaxation term towards a desired velocity function
Since we are considering identical vehicles we assume
The model is given by
{˙xn=vn˙vn=α(V(Δxn)−vn)+βΔvn(Δxn)2 | (4) |
with
In our study we usually refer to a circular road which means to solve 4 with periodic boundary conditions, in this way the vehicle with index
Let us characterize the equilibrium for the single-lane model.
Proposition 1. The equilibrium of the system 4 is given if all vehicles are equally spaced and move with the same constant velocity.
In fact, let us indicate with
{xn+1(0)−xn(0)=hvn(0)=V(h)for n=1,…,N | (5) |
with
ˉxn(t)=hn+V(h)t. | (6) |
Note that the equation depends parametrically by the given number
Now we study the stability of model 4 around the equilibrium 6 by linearizing the original system. Let
xn=ˉxn+yn. | (7) |
Disregarding terms higher than
¨yn=α(V′(h)Δyn−˙yn)+βΔ˙ynh2 | (8) |
where
We solve 8 looking for solutions
yk(n,t)=exp{iakn+zt} | (9) |
where
z2+z(α−βh2(eiak−1))−αV′(h)(eiak−1)=0. | (10) |
If the amplitude of
Let us write the two solutions of 10 as
ℜ(z1+z2)=u1+u2=−α+βh2(cos(ak)−1)ℑ(z1+z2)=v1+v2=βh2sin(ak)ℜ(z1⋅z2)=u1⋅u2−v1⋅v2=−αV′(h)(cos(ak)−1)ℑ(z1⋅z2)=u1⋅v2−v1⋅u2=−αV′(h)sin(ak). |
The boundary of the stability region is obtained when
v1=−αV′(h)sin(ak)−α+βh2(cos(ak)−1). |
After some algebraic manipulations we get
V′(h)=α2cos2(ak2)+βh2+2tan2(ak2)⋅βh2(βαh2+1). | (11) |
We can study this problem with polar coordinates in the
Thus we have proved the following result.
Proposition 2. If
V′(h)<α2+βh2 | (12) |
the steady state 6 of model 4 is stable, because for all
For
Now we present some numerical tests of model 4 using the Runge Kutta 5 method, with time step
Let us fix
V(Δx)=max{0,VHT(Δx)} | (13) |
see Fig 3, where
VHT(Δx)=V1+V2tanh(C1(Δx−lc)−C2) | (14) |
is the function given by Helbing and Tilch in [13] where they carried out a calibration of model 3 respect to the empirical data, obtaining the optimal parameter values
From condition 12 we obtain that the model 4 with velocity 13 is stable if
In the next two simulations we show a comparison between model 4 and model 3, perturbing the system adding or removing a vehicle. The initial number of vehicles is chosen in such a way that the model 4 is stable while the model 3 is unstable according their stability condition.
In this simulation we consider
Model 4:
Model 3:
We can see how the perturbation is absorbed in the in first model while it causes a creation of stop & go waves in the second model.
In this simulation we consider again
Model 4:
Model 3:
Also in this test we can observe the differences when a perturbation occurs in the two models.
In this simulation we start with
Model 4:
Model 3:
An example of instability for both models is reported. Although stop & go waves occur we can appreciate the differences in the oscillations of the velocity in the two models and the lack of region with zero speed in model 4.
Here we study the extension of model 4 to a road with two lanes, where lane changing is allowed: lane 1 is the driving lane, while lane 2 is the fast lane. We consider a single population of homogeneous vehicles and we assume that the coefficients
Let
Assuming that vehicle
The model can be written for
{˙xn=vn˙vn=α(Vj(Δxjn)−vn)+βΔvjn(Δxjn)2n∈Ij + lane changing conditions | (15) |
where
V1(Δx)=V2(Δx)=0Δx⩽ds (security distance)V1(Δx)⩽V2(Δx)otherwise. | (16) |
The parameter
The lane changing rules are based according essentially on two criteria: a vehicle may change lane if it would travel at a faster speed in the new lane, which means that is would have a higher acceleration (incentive criterion); and the changing action must be safe in order to avoid collisions with the vehicles in the adjacent lane, which means to held the security distance in every movement (security criterion).
For simplicity we introduce the compact notations:
d(n,m)=xm−xn,aj(n,m)=α(Vj(d(n,m))−vn)+βvm−vn(d(n,m))2 | (17) |
to denote the difference of positions between vehicles with indices
Thus the lane changing rules from lane
aj′(n,sj′n)>aj(n,sjn)(incentive criterion)d(n,sj′n)>dsandd(pj′n,n)>ds(security criterion) | (18) |
In particular cases we have:
● if
● if
● if
● if
Note that in this model lane changes are instantaneous and the velocity of the vehicle remains the same after the changing action. The vehicles following in the new lane adjust their velocities according to the distance from the new vehicle.
In order to reproduce a realistic description of traffic flow, we introduce a physical timer for lane changing because, as reported by experimental studies [18], lane changing is not frequent. In other words, although a vehicle might have the opportunity and the advantage in changing lane, most often drivers prefer not to change lane. Therefore we set an expected number of lane changes per second
In the following we will use to this characterization of a steady state of model 15.
Proposition 3. A steady state of model 15 is obtained when both lanes are in equilibrium and there are no lane changing. The equilibrium velocity is given by the optimal velocity functions.
It is easy to show that such steady state for the two-lanes model 15 is given when the vehicles moves with the same uniform headways
V1(h1)=V2(h2). | (19) |
Recalling that
h2=Lh1Nh1−L | (20) |
and if the equilibrium velocity is less than
Veq:=V1(ˉh1)=V2(ˉh2). | (21) |
Now we prove that if 21 holds we have no lane changes and both lanes remain at equilibrium. Consider model 15 with
∀n∈I1{xn(0)equally spaced with distanceˉh1vn(0)=Veq∀n∈I2{xn(0)equally spaced with distanceˉh2vn(0)=Veq. | (22) |
For the lane change from lane
a2(n,s2n)>a1(n,s1n) | (23) |
is never verified because
a2(n,s2n)−a1(n,s1n)==α(V2(xs2n−xn)−vn)+βvs2n−vn(xs2n−xn)2+X−α(V1(xs1n−xn)−vn)−βvs1n−vn(xs1n−xn)2=V2(xs2n−xn)−V1(xs1n−xn). | (24) |
Moreover
We have proved the following result.
Proposition 4. Consider the system 15 with initial conditions 21-22, then no lane changing occurs.
In the following we study the stability of this equilibrium solution perturbing the initial headways in a lane and analysing the possibility of lane changing in both lanes. We start perturbing the slow lane (lane 1) and then the fast lane (lane 2). Thus we start from an initial condition in which lane 1 is in a local equilibrium but does not satisfy the global equilibrium we described above. This means that we consider a uniform perturbation
We study the possibility of lane changes from lane 1 to lane 2. Let us consider a vehicle with index
a2(n,s2n)?>a1(n,s1n)⇔V2(d2)−V1(ˉh1+ε)+γd22(V2(ˉh2)−V1(¯h1+ε))?>0 | (25) |
where
Consider now the case
ε<V−11(V2(d2)+γd22V2(ˉh2)1+γd22)−ˉh1. | (26) |
Recalling the security criterion we have that an admissible distance
ε<V−11(V2(ˉh2−ds)+γ(ˉh2−ds)2V2(ˉh2)1+γ(ˉh2−ds)2)−ˉh1<0. | (27) |
Using a Taylor expansion for
V2(d2)−V2(ˉh2)−ε(1+γd22)V′1(ˉh1)?>0 | (28) |
is satisfied provided
ε<V2(d2)−V2(ˉh2)(1+γd22)V′1(ˉh1). | (29) |
Then using the monotonicity of the velocity function we get this a priori bound, approximated at the first order respect to
ε<V2(ˉh2−ds)−V2(ˉh2)(1+γ(ˉh2−ds)2)V′1(ˉh1)<0. | (30) |
So if
Consider a vehicle with index
a1(n,s1n)?>a2(n,s2n). | (31) |
In this case clearly we will not have lane changes if
V1(d1)−V1(ˉh1)+γd21V′1(ˉh1)ε?>0 | (32) |
where
ε>ds. | (33) |
In fact the arrival of a vehicle from lane 2 modifies the initial perturbation
Now we repeat the same analysis adding a perturbation
In this case a vehicle in lane 1 could clearly have a greater acceleration from lane 1 to the lane perturbed if
ε>ds | (34) |
as seen in case 2.
If the perturbation
a1(n,s1n)?>a1(n,s2n)⇔V1(d1)−V2(ˉh2+ε)+γd21(V1(ˉh1)−V2(ˉh2)+ε))?>0 | (35) |
with admissible distance
ε<V−12(V1(d1)+γ(d1)2V1(ˉh1)1+γ(d1)2)−ˉh2<0 | (36) |
which can be linear approximated by
ε<V1(d1)−V1(ˉh1)(1+γd21)V′2(ˉh2). | (37) |
Then using the monotonicity of the velocity function we get this a priori bound, approximated at the first order respect to
ε<V1(ˉh1−ds)−V1(ˉh1)(1+γ(ˉh1−ds)2)V′2(ˉh2)<0. | (38) |
We can summarize the results in the following proposition.
Proposition 5. Starting from the equilibrium, lane changing for system 15 are activated if a perturbation
Thresholds and perturbations
.from lane 1 to lane 2 | from lane 2 to lane 1 | |
perturbation |
||
perturbation |
Here we present some numerical tests for the two-lane model 15, using the Runge Kutta 5 method. In the following simulations we set a maximum number of lane changes per second equal to
Let us set
V1(h)={5tanh(0.02(h−5))if h>ds0otherwiseV2(h)=2V1(h). | (39) |
with
In this simulation we want to study the perturbation of the lane one from the equilibrium state. Let us fix
Now we want to perturb the lane 1 adding new vehicles. From bound 30 we obtain that the perturbation
Thus fix
∀n∈I1{xn(0)equally spaced with distanceˉh1+˜εvn(0)=V1(ˉh1+˜ε)∀n∈I2{xn(0)equally spaced with distanceˉh2vn(0)=V1(ˉh2) | (40) |
Fig. 15 shows the simulation for
Whit this simulation we want to study the possibility of lane changes from lane 2 to lane 1. We consider again the equilibrium found in Test 1, and we focus attention to perturb the headways in lane 1 with a positive value of
From 33 we know that a perturbation that activates lane changes from lane 2 to lane 1 must be greater that the security distance. In our case this is verify if we consider
∀n∈I1{xn(0)equally spaced with distanceˉh1+˜εvn(0)=V1(ˉh1+˜ε)∀n∈I2{xn(0)equally spaced with distanceˉh2vn(0)=V1(ˉh2) | (41) |
Fig. 16 shows the simulation for
In this simulation we study the evolution towards equilibrium. We start with the same number of vehicles in both lanes
Fig. 17 shows the simulation for
In this simulation we use the two velocity functions as in 13 in order to consider also the instability due to the number of vehicles as seen in the single-lane case. We fix
V1(Δx)={6.75+7.91tanh(0.13(Δx−5)−1.57)Δx>50otherwiseV2(Δx)=2V1(Δx). |
The stability conditions for the single-lane 12 are in this case: for lane 1 stability for
We start with the same number of vehicles in both lanes
∀n∈I1{xn(0)−xn−1(0)=LN1(0)+rnvn(0)=V1(LN1(0))∀n∈I2{xn(0)−xn−1(0)=LN2(0)vn(0)=V2(LN2(0)) | (42) |
Fig. 18 shows the simulation for
The model 15 can be easily generalized to the multi-lane case with a generic number of lanes. We can differentiate the lanes by attributing different profiles of desired velocity, therefore let
The model can be written as
{˙xn=vn˙vn=α(Vj(Δxjn)−vn)+βΔvjn(Δxjn)2n∈Ij + lane changing conditionsfor j=1,…,J. | (43) |
We adopt the lane changing conditions as in 18. Note that, except for the cases
As we done for the two-lane model we can define the steady state of model 43 in which all lane are at the equilibrium and lane changes do not occur. This is provided for the values of the headways
ˉh1,…,ˉhJ | (44) |
that verify the condition
V1(ˉh1)=⋯=VJ(ˉhJ). | (45) |
In order to find this equilibrium we require also that the equilibrium velocity defined in 45 must be smaller than the value
Let us consider a three-lane road (
We are now interested to add a perturbation in the middle lane and to study the possibility of lane changing. More specifically let us consider the initial conditions
∀n∈I1{xn(0)equally spaced with distanceˉh1vn(0)=Veq∀n∈I2{xn(0)equally spaced with distanceˉh2+εvn(0)=V2(ˉh2+ε)∀n∈I3{xn(0)equally spaced with distanceˉh3vn(0)=Veq. | (46) |
We can observe that the system is comparable to two subsystems: lane 1 - lane 2 and lane 2 - lane 3 where the lane changes are regulated by the thresholds in Table 1. More specifically for the subsystem lane 2 - lane 3 we consider the case of a perturbation in the slow lane (first row of the table) while for the subsystem lane 1 - lane 2 we refer to the case of a perturbation in the fast lane (second row of the table). We add to this framework the possibility of choosing the best advantageous change for a vehicle in the middle lane that might have two possibilities for change lane. The thresholds that enable lane changes can be obtained from Table 1 with the appropriate modifications. We have
Here we propose a numerical example with a three-lane road, using the Runge Kutta 5 method. Consider the velocity function
Thresholds and perturbations
.pert. |
In the following test we show an example of instability, comparing the results with the test in section 3.3.4. Let us consider the function
∀n∈I1{xn(0)−xn−1(0)=LN1(0)+rnvn(0)=V1(LN1(0))∀n∈I3{xn(0)−xn−1(0)=LN3(0)vn(0)=V2(LN3(0)) | (47) |
From Fig. 22 we can see that lane 1 gradually empties into lane 2. Due to frequent lane changes, more pronounced stop & go waves occur in fast lanes, while slow lane tends to stabilize. In test 3.3.4 we recall that the instabilities were evident in both lanes.
In this paper we have studied a microscopic model 4 for lane changing proposing simple lane changing rules. We have computed global steady states and we have investigated the linear stability of such solutions. The global steady state of the multi-lane model is parametrized by the total number
Both authors are members of the INdAM Research group GNCS. This work was supported in part by Progetto di Ateneo 2019, n. 1622397 and 2020 n. 2023082 (Sapienza - Università di Roma), and PRIN 2017KKJP4X.
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Thresholds and perturbations
.from lane 1 to lane 2 | from lane 2 to lane 1 | |
perturbation |
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perturbation |
Thresholds and perturbations
.pert. |
from lane 1 to lane 2 | from lane 2 to lane 1 | |
perturbation |
||
perturbation |
pert. |