Discrete velocity models and relaxation schemes for
traffic flows
M. Herty∗
L. Pareschi†
M. Seaı̈d‡
Abstract
We present simple discrete velocity models for traffic flows. The novel
feature in the corresponding relaxation system is the presence of non negative velocities only. We show that in the small relaxation limit the discrete
models reduce to the Lighthill-Whitham-Richards equation. In addition
we propose second order schemes combined with IMEX time integrators
as proper discretization of the relaxation-type system. Numerical tests
are carried out on various situations in traffic flow. The results show that
the proposed models are capable to describe correctly the formation of
backward waves induced by traffic jam.
Keywords. Discrete velocity models, Relaxation schemes, IMEX schemes,
ENO schemes, Lighthill-Whitham-Richards equation.
Contents
1 Introduction
2
2 Discrete velocity models of traffic flows
4
2.1
Two speeds models . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Three speeds models . . . . . . . . . . . . . . . . . . . . . . . . .
6
∗
FB Mathematik, TU Darmstadt, 64289 Germany ([email protected])
Department of Mathematics, University of Ferrara, 44100 Italy ([email protected])
‡
FB Mathematik, TU Darmstadt, 64289 Germany ([email protected])
†
1
3 Relaxation schemes
8
3.1
Space discretization . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3
Relaxed schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Numerical examples
8
12
4.1
Free-flow traffic situation
. . . . . . . . . . . . . . . . . . . . . . 12
4.2
Jam situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.3
Bottleneck situation . . . . . . . . . . . . . . . . . . . . . . . . . 13
5 Conclusions
13
References
15
1
Introduction
Macroscopic modeling of vehicular traffic started with the work of Lighthill and
Whitham [12]. They considered the continuity equation for the density ρ and
approximated the mean velocity u by an equilibrium value ue (ρ)
∂t ρ + ∂x (ρue (ρ)) = 0.
(1)
The function ue (ρ) is also called fundamental diagram and ρue (ρ) is the flux.
For a comparison with experimental data we refer to [13]. Including additional
momentum equations for u, we can derive so called “higher order” models for
traffic flow. Examples and mathematical studies of “second order” models can
be found in [19, 3, 1].
Another type of mathematical models used in traffic dynamics are kinetic models. Therein, a kinetic car density f is introduced. Kinetic equations can be
found, for example, in [20, 21, 9, 4]. Procedures to derive macroscopic traffic
equations from underlying kinetic models have been performed in different ways
by various authors, see [18, 9].
In this paper we focus on simple discrete velocity models of traffic flows which
share the common properties that in the small relaxation limit reduce to a
macroscopic description of traffic flow by the single conservation law (1).
2
In [7], Jin and Xin proposed the relaxation system
∂t ρ + ∂x q = 0,
∂t q + a2 ∂x ρ = −
1
q − F (ρ) ,
ε
(2)
as an approximation to the scalar hyperbolic equation
∂t ρ + ∂x F (ρ) = 0.
(3)
Formally, solution of the relaxation system (2) approximates solution to the
original equation (3) by order O(ε) if the constant a satisfies the subcharacteristic condition [7, 6, 2, 14]
a2 − F ′ (ρ)2 ≥ 0,
∀ ρ.
(4)
At a numerical level, the main advantage in considering a relaxation system
is the fact that the nonlinear conservation law (3) is replaced by a semi-linear
system (2) with linear characteristic variables given by v ± au. Consequently
nonlinear Riemann problems, characteristic decompositions and linear iterations are avoided in its numerical solution. The idea of developing relaxation
methods to approximate numerical solutions to partial differential equations
has a long tradition. This field of research is very active for hyperbolic systems
of conservation laws, where a vast number of relaxation schemes have been designed based on high-order reconstructions and shock capturing techniques. We
refer the reader to [10, 7, 5, 22] and further references can be found therein. All
of these methods are easy to formulate and to implement. Moreover, there is
strong links between relaxation methods and central schemes, see for instance
[24, 16].
The relaxation system (2) is strictly connected to discrete velocity models in
kinetic theory. In fact it can be reformulated by considering “kinetic” variables
f and g as follows
1
1
q
∂t f + a∂x f = −
,
f−
ρ+
ε
2
a
(5)
1
1
q
∂t g − a∂x g = −
g−
ρ−
,
ε
2
a
where ρ = f + g and q = a(f − g). The equations (5) can be viewed as an
evolution system of particles with speeds ±a and interaction terms given by
the right-hand side, compare [15, 8, 17].
Note that in (5) the kinetic variables f and g propagate with positive and
negative speed, respectively. This fits well with the fact that gas particles can
move in either direction. However, if we restrict to a situation where all particles
have the same (nonnegative) propagation direction, then the above formulation
will not describe the correct dynamics of the relaxed flow. If we consider the
3
case of particles travelling with speeds b > a ≥ 0 easy computations lead to the
subcharacteristic condition
F ′ (ρ) − a b − F ′ (ρ) ≥ 0,
(6)
which is not satisfied unless a ≤ F ′ (ρ) ≤ b. A physical field where such situation
takes place is the case of traffic flow models where cars can have only positive
velocities but a traffic jam moves also backwards. Therefore, it is our goal here
to derive simple relaxation-type models with a clear “kinetic” interpretation
since we allow car to propagate only with nonnegative speeds but still able to
capture the backward propagation of the solution.
Our objective in this paper is twofold: on the one hand, we develop simple
discrete velocity models for traffic flow problems, on the other hand we develop
relaxation schemes which are able to capture the correct asymptotic limit even
when only nonnegative positive speeds are used. The accuracy of the method is
illustrated by numerical examples for traffic flow in presence of both smooth and
shock solutions. In the following section, we present the traffic flow equations
used in the paper to develop relaxation-type model. In section 3, the relaxation
scheme for the model problem is detailed. This includes the space and time
discretizations. The numerical results are presented in section 4. We apply
our scheme to common situations like free flow, bottleneck and traffic jam. In
section 5, some concluding remarks are listed.
2
Discrete velocity models of traffic flows
In traffic flow theory the time evolution of a vehicle density in a freeway is
governed by the Lighthill-Whitham-Richards model [12]
∂t ρ + ∂x V (ρ) = 0,
(7)
where ρ(t, x) is the vehicle density at location x and time t, and V (ρ) is the
flow function given by
V (ρ) = ρ min{u0 , ue (ρ)},
(8)
where ue (ρ) is the fundamental diagram, i.e. an equilibrium velocity, and u0
the minimum speed. A typical choice for ue (ρ) is for example
ρ
,
(9)
ue (ρ) = um 1 −
ρm
with ρm and um are the maximum density and the maximum speed, respectively.
Remark 1 More general flux functions as well as general traffic flow models
have been considered in the literature, see [9, 1]. For inhomogeneous traffic flow
situations we may have an additional dependence on x, i.e. V (ρ) = V (x, ρ).
4
2.1
Two speeds models
Let us consider the following simple discrete velocity model
∂ t f + v1 ∂ x f
= −
∂ t g + v2 ∂ x g =
K(ρ) v1 f + v2 g − V (ρ) ,
2ε
K(ρ) v1 f + v2 g − V (ρ) ,
2ε
(10)
(11)
where K(ρ) is a suitable function such that

K(ρ) ≥ 0, if ρ ≤ ρ∗ ,
K(ρ) < 0, if
ρ > ρ∗ ,
with 0 < ρ∗ < ρm is a constant.
The system (10)-(11) offers the following interpretation. We have two species
of cars driving with velocity v2 > v1 ≥ 0. The functions f and g are probability
distributions of cars with speeds v1 and v2 , respectively. Depending on the
density, K(ρ) changes sign. Usually in traffic flow models, the flux function
V : ρ → ρue (ρ) is a concave function. Hence K(ρ) ≥ 0 in low density regimes,
whereas K(ρ) should become large and negative in high density regimes so that
there is gain for f and a loss for g by the interaction terms. Therefore, more
cars will attain the speed v1 . This fits well to the observation, that in areas
of high densities the cars tend to slow down. The opposite phenomena is also
governed by the equations (10)-(11).
In the sequel we assume that
K(ρ) = H V ′ (ρ) − v1 ,
(12)
where H(·) is a function that preserves the sign of the argument (for example
a Heavyside or a hyperbolic tangent function, see Figure 1).
Introducing the density and the flux of cars defined as
ρ=f +g
and
J = v1 f + v2 g,
(13)
we obtain the following macroscopic system
∂t ρ + ∂x J
= 0,
∂t J + (v1 + v2 )∂x J − v1 v2 ∂x ρ = −
(14)
K(ρ)
(v2 − v1 ) J − V (ρ) .
ε
(15)
In the small relaxation limit (ε → 0), the equation (15) gives to leading order
ε
V ′ (ρ) − v1 V ′ (ρ) − v2 ∂x ρ .
J = V (ρ) +
K(ρ) (v2 − v1 )
5
By using this variable in the equation (14) we obtain
∂t ρ + ∂x V (ρ) + ∂x
ε
V ′ (ρ) − v1 V ′ (ρ) − v2 ∂x ρ
K(ρ) (v2 − v1 )
!
= 0. (16)
To ensure the dissipative nature of (16), it is necessary that
(V ′ (ρ) − v1 ) (V ′ (ρ) − v2 )
≤ 0.
K(ρ) (v2 − v1 )
(17)
This condition is satisfied by the assumption (12) on the kernel K(ρ) if v1 ≤
um ≤ v2 . Formally, when ε → 0, the relaxation system (14)-(15) converges
to the original equation (7) provided the subcharacteristic condition (17) is
satisfied.
Note that the equilibrium states for f and g are defined by
ρ
V (ρ)
V (ρ)
ρ
Ef (ρ) =
v2 −
− v1 .
,
Eg (ρ) =
v2 − v1
ρ
v2 − v1
ρ
(18)
Nonegativity of the equilibrium states is guaranteed if
v1 ≤
V (ρ)
≤ v2 ,
ρ
(19)
which is satisfied under the natural requirements u0 ≥ v1 and um ≤ v2 .
Remark 2 It is remarkable that when the density in the system becomes critical, i.e. ρ → ρm , a desirable feature would be that all cars reduce their speed
to the lowest speed v1 without allowing f + g > ρm . This behavior is obtained
naturally for small values of ε in agreement with the kinetic interpretation of ε
as the mean free path between car-particles.
As an alternative, in order to have this feature even for large values of ε we can
ask that K(ρ) → −∞ as ρ → ρm to recover the critical equilibrium states
f=
ρm
(v2 − u0 ) ,
v2 − v1
g=
ρm
(u0 − v1 ) ,
v2 − v1
(20)
which correspond to a traffic jam situation. Clearly taking u0 = v1 we have
f = ρm and g = 0.
2.2
Three speeds models
In this section we extend the previous idea by considering models with three
different speeds v2 , v1 , 0 s.t. v2 > v1 > 0. The model describing the traffic
interaction will be given by
6
∂ t f + v1 ∂ x f
K(ρ)(v2 − v1 ) Ef (ρ) − f ,
2ε
=
K(ρ)(v2 − v1 ) Eg (ρ) − g ,
2ε
K(ρ)(v2 − v1 ) Eh (ρ) − h ,
2ε
∂ t g + v2 ∂ x g =
∂t h =
where the equilibrium states are
V (ρ) um − v1
V (ρ) v2 − um
, Eg (ρ) =
,
Ef (ρ) =
um
v2 − v1
um
v2 − v 1
Eh (ρ) = ρ −
(21)
(22)
(23)
V (ρ)
.
um
In (21)-(23), K(ρ) is a kernel as in the two speeds model. Nonnegativity of the
equilibrium states is guaranteed under the natural assumption v2 ≥ um ≥ v1 .
Introducing the macroscopic variables
ρ = f + g + h,
J = v1 f + v2 g,
z = f + g,
(24)
we obtain the following set of equations
∂ t ρ + ∂x J
= 0,
K(ρ)
(v2 − v1 ) J − V (ρ) ,
ε
V (ρ) K(ρ)
(v2 − v1 ) z −
.
= −
ε
um
∂t J + (v1 + v2 )∂x J − v1 v2 ∂x z = −
∂t z + ∂x J
(25)
(26)
(27)
In the small relaxation limit (ε → 0), equation (27) gives z = V (ρ)/um and
thus equation (26) gives to leading order
v1 v2
ε
′
′
V (ρ) − (v1 + v2 ) +
J = V (ρ) +
V (ρ)∂x ρ .
K(ρ) (v2 − v1 )
um
By using this variable in the equation (25) we obtain
∂t ρ+∂x V (ρ)+∂x
εV ′ (ρ)
K(ρ) (v2 − v1 )
!
v1 v 2
′
∂x ρ = 0. (28)
V (ρ) − (v1 + v2 ) +
um
To ensure the dissipative nature of (28), it is necessary that
v1 v2
V ′ (ρ)
V ′ (ρ) − (v1 + v2 ) +
≤ 0.
K(ρ)(v2 − v1 )
um
(29)
The condition is satisfied by assuming
K(ρ) = H V ′ (ρ) ,
where H(·) is a function that preserves the sign of the argument.
7
(30)
Remark 3 Again, when the density in the system becomes critical, i.e. ρ →
ρm , if we ask that K(ρ) → −∞ we obtain the critical equilibrium states
u 0 v2 − u m
u 0 u m − v1
u0
Ef (ρ) =
, Eg (ρ) =
, Eh (ρ) = ρ −
,
u m v2 − v1
u m v2 − v1
um
corresponding to the traffic jam situation. Clearly now we can take u0 = 0 to
get f = 0, g = 0 and h = ρm .
3
Relaxation schemes
We present a numerical method to approximate the discrete velocity models
in such a way that the resulting scheme is asymptotic preserving (AP) and
high order accurate in space and time (we refer to [16] for more details on AP
schemes).
The numerical method we construct is based on the method of lines approach.
We use a high order central WENO scheme as space discretization of the kinetic
system. The necessity to use a central scheme comes from the fact that the
limiting equation (7) allows solutions ρ(t, x) which can propagate with either
positive or negative speed (see the Appendix). For the time integration we
apply a third-order IMEX Runge-Kutta method.
For simplicity we will describe the numerical schemes for the two speeds discrete
velocity model. The extension to the three speeds model is straightforward.
3.1
Space discretization
Let the spatial interval discretized in equally spaced grids with stepsize ∆x,
and let wi denotes the point-value of an arbitrary function w at gridpoint xi .
Then a semi-discretization of the equation (10) can be written as
fi+ 1 − fi− 1
dfi
2
2
+ v1
= G+
i .
dt
∆x
(31)
For simplicity in presentation and in order to show how a space discretication
can be reconstructed we consider only the Lax-Friedrichs method. Extension to
higher order reconstructions can be developed using similar formulation. Thus,
the first order Lax-Friedrichs numerical flux fi+ 1 is given by
2
fi+ 1 =
2
1
2
∆x
fi+1 + fi −
(fi+1 − fi ) .
v1 ∆t
8
(32)
Similarly, the semi-discretization of equation (11) can be written as
gi+ 1 − gi− 1
dgi
2
2
+ v2
= G−
i .
dt
∆x
(33)
where gi+ 1 is defined analogously by
2
gi+ 1
2
1
=
2
∆x
(gi+1 − gi ) .
gi+1 + gi −
v2 ∆t
(34)
The source terms G±
i are given by
G±
i =±
K(ρi ) v1 fi + v2 gi − V (ρi ) .
2ε
(35)
dρi Ji+ 21 − Ji− 12
+
= 0,
dt
∆x
(36)
Using (18) in (31) and (33) gives a first-order semi-discretization of the relaxation system (14)-(15)
Ji+ 1 − Ji− 1
ρi+ 1 − ρi− 1
dJi
2
2
2
2
+ (v1 + v2 )
− v 1 v2
=
dt
∆x
∆x
−
where
ρi+ 1 = fi+ 1 + gi+ 1
2
2
2
and
K(ρi )
(v2 − v1 ) Ji − V (ρi ) ,
2ε
(37)
Ji+ 1 = v1 fi+ 1 + v2 gi+ 1 ,
2
2
2
with fi+ 1 and gi+ 1 are given by (32) and (34), respectively.
2
2
Higher order spatial discretizations can be derived following the ENO reconstructions, compare [23, 24, 11] among others. For instance, a second order
reconstruction is obtained by substituting the numerical fluxes fi+ 1 and gi+ 1
2
2
using the convex ENO scheme [11]. Thus,
and
gi+ 1 = R v2 , gi+ 1 ,
(38)
fi+ 1 = R v1 , fi+ 1
2
2
2
2
where the reconstruction flux function R is defined as
1
∆x
R α, wi+ 1
=
(wi+1 − wi ) +
wi+1 + wi −
2
2
α∆t
∆x
1
α
wi+1 − wi +
(wi+1 − wi ) φ θ+
−
4
α∆t
1
∆x
α
wi+2 − wi+1 −
(wi+2 − wi+1 ) φ θ−
,
4
α∆t
(39)
α and θ α are given by
with the slopes θ+
−
α
θ+
=
wi − wi−1 +
wi+1 − wi +
∆x
α∆t
∆x
α∆t
(wi − wi−1 )
(wi+1 − wi )
,
α
θ−
=
9
wi+1 − wi −
wi+2 − wi+1 −
∆x
α∆t
∆x
α∆t
(wi+1 − wi )
(wi+2 − wi+1 )
,
and φ is a slope limiter function. A simple choice is the minmod limiter
φ(θ) = max (0, min (1, θ)) .
Note that if we set φ = 0 in (39), the reconstruction (38) reduces to the first
order Lax-Friedrichs discretization (32) and (34). We should mention that any
discretization that requires characteristics information will generate instabilities
in the computed solution. This is due to the structure of the relaxation system
which allows propagation along the characteristics in one direction only.
3.2
Time integration
The fully discretization of system (14)-(15) can be obtained by the well established IMEX methods, see for instance [16, 10]. The special structure of the
nonlinear terms in (37) makes it trivial to evolve the flux terms explicitly and
the source term implicitly. Denote by
vi+ 1 − vi− 1
2
.
∆x
With ∆t being the time step and win denotes the approximate of a function
w at time t = n∆t and gridpoint xi , the implementation of the third order
Runge-Kutta method to solve (36)-(37) can be carried out in the following
steps:
D x vi =
2
ρ∗i = ρni ,
Ji∗ = Jin − 3
(1)
ρi
(1)
Ji
∆t
K(ρ∗i ) (v2 − v1 ) (Ji∗ − V (ρ∗i )) −
2ε
∆t
K(ρni ) (v2 − v1 ) (Jin − V (ρni )) ,
2ε
= ρ∗i − ∆tDx Ji∗ ,
= Ji∗ − ∆t(v1 + v2 )Dx Ji∗ + ∆tv1 v2 Dx ρ∗i ,
(1)
= ρi ,
ρ∗∗
i
(1)
Ji∗∗ = Ji
+2
∆t
∗∗
∗∗
K(ρ∗∗
i ) (v2 − v1 ) (Ji − V (ρi )) +
2ε
∆t
K(ρ∗i ) (v2 − v1 ) (Ji∗ − V (ρ∗i )) −
2ε
∆t
K(ρni ) (v2 − v1 ) (Jin − V (ρni )) ,
2ε
3 n 1 ∗∗
ρ + (ρi − ∆tDx Ji∗∗ ) ,
4 i
4
3 n 1 ∗∗
J + (Ji ∆t(v1 + v2 )Dx Ji∗∗ + ∆tv1 v2 Dx ρ∗∗
i ),
4 i
4
(2)
ρi ,
∆t
(2)
∗∗∗
− V (ρ∗∗∗
K(ρ∗∗∗
Ji −
i )) −
i ) (v2 − v1 ) (Ji
2ε
3∆t
∗∗
∗∗
K(ρ∗∗
i ) (v2 − v1 ) (Ji − V (ρi )) −
8ε
10
(2)
=
(2)
=
ρi
Ji
ρ∗∗∗
=
i
Ji∗∗∗ =
10
∆t
2∆t
K(ρ∗i ) (v2 − v1 ) (Ji∗ − V (ρ∗i )) +
K(ρni ) (v2 − v1 ) (Jin − V (ρni )) ,
2ε
4ε
1 n 2 ∗∗∗
ρ + (ρi − ∆tDx Ji∗∗∗ ) ,
ρn+1
=
i
3 i
3
1 n 2 ∗∗∗
n+1
J + (Ji − ∆t(v1 + v2 )Dx Ji∗∗∗ + ∆tv1 v2 Dx ρ∗∗∗
Ji
=
i ).
3 i
3
Note that in the above IMEX scheme neither linear algebraic equations no
nonlinear source terms can arise. Furthermore, the obtained relaxation scheme
is stable independently of ε such that the selection of ∆t is based only on the
usual CFL condition
∆t
CFL = max {v1 , v2 }
≤ 1.
(40)
∆x
3.3
Relaxed schemes
Using a Hilbert expansion for (36)-(37) and neglecting terms of higher order in
ε we obtain
Ji = V (ρi ),
and
1
∆x
V (ρi+1 ) + V (ρi ) −
(V (ρi+1 ) − V (ρi )) .
2
2
∆t
This yields a first order Lax-Friedrichs discretization of the flux for the semidiscrete scheme
dρi Ji+ 21 − Ji− 12
+
= 0.
(41)
dt
∆x
Similarly, when ε −→ 0, the reconstruction (38) yields to the second order
convex ENO scheme for the relaxed equation (41) with
1
∆x
V (ρi+1 ) + V (ρi ) −
(V (ρi+1 ) − V (ρi )) +
Ji+ 1 =
2
2
∆t
1
∆x
(V (ρi+1 ) − V (ρi )) φ (θ+ ) −
V (ρi+1 ) − V (ρi ) +
4
∆t
1
∆x
(V (ρi+2 ) − V (ρi+1 )) φ (θ− ) ,
V (ρi+2 ) − V (ρi+1 ) −
4
∆t
Ji+ 1 =
where
θ+ =
θ− =
∆x
∆t (V (ρi ) − V (ρi−1 ))
,
∆x
(ρi+1 ) − V (ρi ) + ∆t (V (ρi+1 ) − V (ρi ))
V (ρi+1 ) − V (ρi ) − ∆x
∆t (V (ρi+1 ) − V (ρi ))
.
∆x
(ρi+2 ) − V (ρi+1 ) − ∆t (V (ρi+2 ) − V (ρi+1 ))
V (ρi ) − V (ρi−1 ) +
V
V
For equilibrium initial data the leading behavior of relaxation scheme is governed by the associated relaxed scheme. Hence, one can recover all physical
properties of the full conservation law (7) by our relaxation-type system with
positive velocities (10)-(11). In addition, at the limit when ε → 0, the time
integration converges to an explicit TVD Runge-Kutta scheme, see [10] for details.
11
4
Numerical examples
In this section we illustrate the performance of our relaxation method for different test cases on traffic flow. Throughout the numerical tests the relaxation
rate ε = 10−1 or ε = 10−10 and the time step ∆t is chosen according to the
CFL condition (40). The kernel K(ρ), and the characteristic speeds v1 and v2
are chosen
v1 = 0.1,
v2 = 1,
(42)
K(ρ) = tanh 5 V ′ (ρ) − v1 ,
for the two speeds model, and
K(ρ) = tanh 5V ′ (ρ) ,
v1 = 0.5,
v2 = 1,
(43)
for the three speeds model. In all the results reported in this section for traffic
flow, the spatial domain is the road interval [0, 1] discretized into 200 gridpoints
and a CFL = 0.5 is used. We display results for the two speeds model at
ε = 10−1 and ε = 10−10 . For the three speeds model, results are shown only
for ε = 10−1 , since for small values of ε these results overlap those obtained by
the two speeds model. The following examples are selected:
4.1
Free-flow traffic situation
We turn to the traffic flow model (7)-(9) in the simplest case where cars moves
freely without jams. Here we assume ρm = um = 1 and initially the cars are
normally distributed around the point x = 0.4 (Gaussian-like initial data). The
propagation of the density in the space-time domain for ε = 10−1 and ε = 10−10
is given in figure 2 whereas figure 3 shows the contour lines of density ρ. Results
obtained by the three speeds model are shown in figure 4 at ε = 10−1 . For
comparison reason, the contour lines of the flux J are also included in figure 2.
As expected, the relaxation scheme evolves the free traffic in the correct way.
Note that because of the different choice of the speeds we made in (42)-(43)
cars have the tendency to drive faster in the three speeds model.
In figure 5 we plot the density at t = 0.6 for different relaxation limits. The
asymptotic convergence of the solution as ε → 0 is clearly preserved by our
discretizations. All these results are computed using the second order ENO
reconstruction.
4.2
Jam situation
Next we consider a similar situation as before but with an additional traffic
jam at x = 0.75. In realistic vehicular traffic flow this is equivalent to relaxed
traffic upstream (x < 0.75) and crowded traffic downstream (x ≥ 0.75). In
12
such situation one expect that the Gaussian-like cars distribution start moving
into the jam, and once they reach the jam a backwards wave should be formed.
By examinating the evolution of the density in figure 6 for ε = 10−10 , we can
observe that the mentioned physical features are well captured by our relaxation
scheme. The contour plots of density ρ and flux J are displayed in figure 7. It
is clear that the Gaussian density distribution is moving into the jam and then
propagating backwards as expected. Figure 9 shows the results for the three
speeds model at ε = 10−1 . As for the two speed model there is an overshoot of
the density at the jam due to the large value of ε. This suggest to take ε small
when the value of the density is close to critical.
All these results are computed using the second order ENO reconstruction. Figure 8 represents a comparison of density plots at t = 0.6 using the first order
Lax-Friedrichs and the second order ENO discretizations. It is clear that excessive diffusion was introduced by the first order Lax-Friedrichs discretization,
while the second order ENO reconstruction sharply captures the shock.
4.3
Bottleneck situation
Our final example is the bottleneck traffic taken from [25]. The problem statement is given by equation (7)-(9) where the flux function V (ρ, x) depends also
on the space coordinate x subject to discontinuous maximum density and velocity
ρ
V (x, ρ) = um (x)ρ 1 −
,
ρm (x)
with
ρm (x) =

4, if

2, if
x < 0.3,
and
x ≥ 0.3,
um (x) =

1,
if
x < 0.3,

0.6, if
x ≥ 0.3.
Initially the road has a constant density ρ(x, 0) = 0.2. The evolution of density
is given in figure 10 and the contour plots of density ρ and flux J are given in
figure 11 at ε = 10−1 and ε = 10−10 . Our relaxation scheme performs well for
this discontinuous situation in traffic flow. In figure 12, we display the results
using the three speeds model at ε = 10−1 . Again the difference in the results
can be explained by the different choice we made for the velocities in (42) and
(43).
5
Conclusions
We have presented a relaxation-type formulation for hyperbolic traffic flow problems. The advection part of these relaxation systems has only nonnegative velocities so that they can be interpreted as simple discrete velocity models for
13
traffic flow problems. We showed that high order central discretizations are an
essential feature in order to simulate also backward traffic waves. The numerical results show that the performance of our second order numerical schemes
is very attractive without solving Riemann problems or requiring special front
tracking.
Acknowledgements: The authors acknowledge the financial support by HYKE
network of the European Union, contract HPRN-CT-2002-00282. M. Herty and
M. Seaı̈d acknowledge the hospitality of the University of Ferrara in Italy where
most of their research was done.
Appendix: Instability of upwind schemes
In this short appendix we show that the most natural approach for the discretization of a system like (10)-(11) becomes unstable in the small relaxation
limit due to the presence of backward waves. Similar arguments applies also to
the three speeds model and can be extended to higher order upwind methods.
The most popular way to solve system (10)-(11) is to consider in two separate
steps an advection problem
∂ t f + v1 ∂ x f
= 0,
(44)
∂t g + v2 ∂x g = 0,
and a stiff relaxation problem
∂t f
= −
∂t g =
K(ρ) v1 f + v2 g − V (ρ) ,
2ε
K(ρ) v1 f + v2 g − V (ρ) .
2ε
(45)
Since the advection part contains only nonnegative speeds we can apply a simple
first order upwinding scheme. If we write the upwind scheme in terms of the
macroscopic variables we get
ρn+1
= ρni −
i
Jin+1
=
Jin
∆t n
n
Ji − Ji−1
,
∆x
∆t
∆t
n
−
(v1 + v2 ) Jin − Ji−1
v1 v2 ρni − ρni−1 .
+
∆x
∆x
(46)
For small values of ε the relaxation step yields the equilibrium projections
V (ρni )
ρni
V (ρni )
ρni
n
n
v2 −
,
gi =
− v1 ,
fi =
(47)
v2 − v1
ρni
v2 − v1
ρni
14
corresponding to Jin = V (ρni ) which substituted into the first equation in (46)
gives
∆t
ρn+1
= ρni −
V (ρni ) − V (ρni−1 ) .
(48)
i
∆x
Clearly (48) is an unstable numerical method for the Lighthill-Whitham-Richards
equation when ρ > ρm /2 and thus V ′ (ρ) < 0.
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15
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16
1
Heaviside(V’(ρ))
tanh(10(V’(ρ)))’
0.8
0.6
0.4
k(ρ)
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.1
0.2
0.3
0.4
0.5
ρ/ρm
0.6
0.7
0.8
0.9
1
Figure 1: Examples of kernels for the discrete velocity models (here v1 = 0 for
the two speeds model).
17
Density ρ
0.8
0.6
0.4
0.2
0
0.9
0.8
1
0.7
0.8
0.6
0.5
0.6
0.4
Space x0.4
Time t
0.3
0.2
0.2
0.1
Density ρ
0.8
0.6
0.4
0.2
0
0.9
0.8
1
0.7
0.8
0.6
0.5
0.6
0.4
Space x0.4
0.3
Time t
0.2
0.2
0.1
Figure 2: Two speeds model: Density evolution for the free-flow traffic situation
at ε = 10−1 (top) and ε = 10−10 (bottom).
18
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
0.8
0.7
0.7
0.6
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.5
0.6
0.7
0.8
0.9
0.1
Time t
0.6
0.7
0.8
0.9
0.6
0.7
0.8
0.9
0.5
0.4
0.4
0.5
Flux J
0.8
0.3
0.4
Density ρ
0.9
0.2
0.3
Time t
0.9
0.1
0.2
Time t
Space x
Space x
Flux J
Space x
Space x
Density ρ
0.2
0.3
0.4
0.5
Time t
Figure 3: Two speeds model: Contour plots of density ρ and flux J for the
free-flow traffic situation at ε = 10−1 (top) and ε = 10−10 (bottom).
19
Density ρ
0.8
0.6
0.4
0.2
0
0.9
0.8
1
0.7
0.8
0.6
0.5
0.6
0.4
Space x0.4
0.2
0.1
Flux J
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Space x
Space x
Density ρ
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.1
0.2
0.3
0.4
0.5
Time t
0.3
0.2
0.6
0.7
0.8
0.9
0.1
Time t
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time t
Figure 4: Three speeds model at ε = 10−1 : Density evolution for the free-flow
traffic situation (top) and contour plots of density ρ and flux J (bottom).
20
0.5
−1
ε = 10
−2
ε = 10
−10
ε = 10
0.45
0.4
0.35
Density ρ
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Space x
Figure 5: Two speeds model: Plots of density for the free-flow traffic situation
at t = 0.6 using different values of ε.
21
Density ρ
0.8
0.6
0.4
0.2
0
0.9
0.8
1
0.7
0.8
0.6
0.5
0.6
0.4
Space x0.4
Time t
0.3
0.2
0.2
0.1
Density ρ
0.8
0.6
0.4
0.2
0
0.9
0.8
1
0.7
0.8
0.6
0.5
0.6
0.4
Space x0.4
0.3
Time t
0.2
0.2
0.1
Figure 6: Two speeds model: Density evolution for the jam traffic situation at
ε = 10−1 (top) and ε = 10−10 (bottom).
22
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
0.8
0.7
0.7
0.6
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.5
0.6
0.7
0.8
0.9
0.1
Time t
0.6
0.7
0.8
0.9
0.6
0.7
0.8
0.9
0.5
0.4
0.4
0.5
Flux J
0.8
0.3
0.4
Density ρ
0.9
0.2
0.3
Time t
0.9
0.1
0.2
Time t
Space x
Space x
Flux J
Space x
Space x
Density ρ
0.2
0.3
0.4
0.5
Time t
Figure 7: Two speeds model: Contour plots of density ρ and flux J for the jam
traffic situation at ε = 10−1 (top) and ε = 10−10 (bottom).
23
Initial
First order LxF
Second order ENO
0.9
0.8
0.7
Density ρ
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Space x
Figure 8: Two speeds model: Plots of density for the jam traffic situation at
t = 0.6 and ε = 10−10 using the first order Lax-Friedrichs and the second order
ENO discretizations.
24
Density ρ
0.8
0.6
0.4
0.2
0
0.9
0.8
1
0.7
0.8
0.6
0.5
0.6
0.4
Space x0.4
0.2
0.1
Flux J
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Space x
Space x
Density ρ
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.1
0.2
0.3
0.4
0.5
Time t
0.3
0.2
0.6
0.7
0.8
0
0.9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time t
Time t
Figure 9: Three speeds model at ε = 10−1 : Density evolution for the jam traffic
situation (top) and contour plots of density ρ and flux J (bottom).
25
0.55
0.5
Density ρ
0.45
1
0.4
0.35
0.8
0.3
0.6
0.25
Time t
0.2
0.4
1
0.8
0.2
0.6
Space x
0.4
0.2
0
0.55
0.5
Density ρ
0.45
1
0.4
0.35
0.8
0.3
0.6
0.25
Time t
0.2
0.4
1
0.8
0.2
0.6
Space x
0.4
0.2
0
Figure 10: Two speeds model: Density evolution for the bottleneck traffic situation at ε = 10−1 (top) and ε = 10−10 (bottom).
26
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
0.8
0.7
0.7
0.6
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.5
0.6
0.7
0.8
0.9
0.1
Time t
0.6
0.7
0.8
0.9
0.6
0.7
0.8
0.9
0.5
0.4
0.4
0.5
Flux J
0.8
0.3
0.4
Density ρ
0.9
0.2
0.3
Time t
0.9
0.1
0.2
Time t
Space x
Space x
Flux J
Space x
Space x
Density ρ
0.2
0.3
0.4
0.5
Time t
Figure 11: Two speeds model: Contour plots of density ρ and flux J for the
bottleneck traffic situation at ε = 10−1 (top) and ε = 10−10 (bottom).
27
0.55
0.5
Density ρ
0.45
1
0.4
0.35
0.8
0.3
0.6
0.25
Time t
0.2
0.4
1
0.8
0.2
0.6
0.4
Space x
0.2
0
Flux J
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Space x
Space x
Density ρ
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
Time t
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time t
Figure 12: Three speeds model at ε = 10−1 : Density evolution for the bottleneck traffic situation (top) and contour plots of density ρ and flux J (bottom).
28