Acta Mathematica Scientia 2009,29B(6):1541–1555

http://actams.wipm.ac.cn

A REACTIVE DYNAMIC CONTINUUM USER

EQUILIBRIUM MODEL FOR BI-DIRECTIONAL

PEDESTRIAN FLOWS∗

Dedicated to Professor James Glimm on the occasion of his 75th birthday

Yanqun Jiang1 Tao Xiong1 S.C. Wong2† Chi-Wang Shu3

Mengping Zhang1 Peng Zhang4 William H.K. Lam5

1.Department of Mathematics, University of Science and Technology of China, Hefei 230026, China

2.Department of Civil Engineering, The University of Hong Kong, Hong Kong, China

3.Division of Applied Mathematics, Brown University, Providence, RI 02912, U.S.A.

4.Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China

5.Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, China

E-mail: jyq2005@mail.ustc.edu.cn; jingt@mail.ustc.edu.cn; hhecwsc@hkucc.hku.hk;

shu@dam.brown.edu; mpzhang@ustc.edu.cn; pengzhang@ustc.edu.cn; cehklam@polyu.edu.hk

Abstract In this paper, a reactive dynamic user equilibrium model is extended to simu-

late two groups of pedestrians traveling on crossing paths in a continuous walking facility.

Each group makes path choices to minimize the travel cost to its destination in a reactive

manner based on instantaneous information. The model consists of a conservation law

equation coupled with an Eikonal-type equation for each group. The velocity-density re-

lationship of pedestrian movement is obtained via an experimental method. The model is

solved using a finite volume method for the conservation law equation and a fast-marching

method for the Eikonal-type equation on unstructured grids. The numerical results ver-

ify the rationality of the model and the validity of the numerical method. Based on this

continuum model, a number of results, e.g., the formation of strips or moving clusters

composed of pedestrians walking to the same destination, are also observed.

Key words pedestrian flows; conservation law; Eikonal-type equation; density-velocity

relationship; finite volume method; fast marching method; unstructured

grids

2000 MR Subject Classification 76M12

∗Received October 22, 2009. The work described in this paper was jointly supported by grants from

the Research Grants Council of the Hong Kong Special Administrative Region, China (HKU 7183/06E), the

University of Hong Kong (10207394) and the National Natural Science Foundation of China (70629001 and

10771134).† Corresponding author: S.C. Wong.

1542 ACTA MATHEMATICA SCIENTIA Vol.29 Ser.B

1 Introduction

Walking is a sustainable mode of transportation that is environmentally friendly, highly

flexible, and healthy, which also serves as an interface with motorized transportation facilities [1,

2]. However, there are inherent dangers associated with large-scale public walking facilities, such

as transfer stations, airports, and train and subway stations. Many tragic crushing incidents

have occurred around the world due to overcrowding in such facilities. For example, in Sheffield,

96 Liverpool fans were killed at the Hillsborough football grounds when police open gates to

alleviate crowding, and the Mena Valley disaster saw 251 Muslims killed in a stampede during

the annual Hajj pilgrimage [3–5]. In most such instances, the fatalities and injuries are caused

by the behavior of the crowd itself, and this issue is receiving increased attention from scientists.

Many crowd-flow models (e.g., [3–19]) have been developed to investigate the complex dynamics

and behavior of pedestrians in various situations.

A number of quantitative observations of pedestrian motion characteristics, including self-

organization phenomena in pedestrian crossing streams and oscillations of the passing direction

at bottlenecks [4, 7–9, 12, 20–22], have been successfully depicted by computer simulation

using micro-simulation models and by empirical or experimental research that makes use of

video analysis. In microscopic models, each individual is represented separately [23, 24]. These

include the cellular automation model [7, 8, 17, 25, 26], the social force mdoel [10, 11, 27,

28], and the lattice gas model [6, 29–31]. Such an approach captures each pedestrian’s unique

situation well, such as his or her route choice, which are particularly well suited for use with

small crowds. Macroscopic models, in contrast, focus on the overall behavior of pedestrian flows

and are more applicable to investigations of extremely large crowds, especially when examining

aspects of motion in which individual differences are less important.

The present research investigates the flux or density distribution and the impact factor

of the speed effect and path choices between two interactive streams of pedestrians, i.e., a bi-

directional pedestrian flow, traveling on crossing paths in a continuous walking facility. Based

on the observation of groups of pedestrians with fluid-like properties moving in a continuous

space, a continuum model is adopted to simulate pedestrian movement. One prior study [15]

provided a systematic framework for a dynamic macroscopic model without considering the

user equilibrium concept. In contrast to the predictive user equilibrium model [13, 14], the

reactive dynamic user equilibrium model describes the movement of pedestrians who do not

have predictive information when they are making a path-choice decision [19, 32]. This means

that pedestrians have to rely on the instantaneous information available to them and make

their choices in a reactive manner to minimize the walking cost to their destination.

The model presented in this paper is designed for bi-directional pedestrian flows and con-

sists of a two-dimensional (2D) scalar hyperbolic conservation law equation coupled with an

Eikonal-type equation for each group. It is extended from a single pedestrian-type reactive

dynamic user equilibrium model [16, 19, 32]. The model system is solved using a finite volume

method (FVM) and a fast marching method (FMM) on an unstructured grid. The former is

simple and capable of flexibly dealing with body boundaries with complicated structures. In

addition, it provides conservative schemes with shock-capturing properties and is extensively

applied to solve the Euler and Navier-Stokes equations (e.g., [33–35]). The FMM, which is

No.6 Y.Q. Jiang et al: A REACTIVE DYNAMIC CONTINUUM USER EQUILIBRIUM MODEL 1543

an ordered upwind method, is appropriate for boundary value problems and is very effective

in solving the continuous Eikonal problem, which is discussed in detail in a number of other

studies [36, 37]. By simulating this continuum model, a number of results, e.g., the formation

of strips or moving clusters composed of pedestrians walking to the same destination, that are

obtained with microscopic models and experimental investigations [22], are also observed here.

The rest of the paper is organized as follows. In Section 2, the governing models of

pedestrian flow are described. The numerical methods for solving the models are given in

Section 3. Section 4 demonstrates a numerical experiment and presents related analysis. Section

5 concludes.

2 Problem Formulation

In this section, a series of formulas for bi-directional pedestrian flows are given. For ease

of reference, the following nomenclature is provided.

(1) Ω represents a 2D continuous walking facility (in m2). (2) Group c represents the c-th

type of pedestrians marching toward the c-th destination of the walking facility. (3) Γ is the

boundary of Ω (in m), Γco the original segment from which Group c enters into the walking

facility (in m), Γcd the destination segment from which Group c leaves the walking facility (in

m), and Γch the wall segment from which nobody in Group c is allowed to enter or leave the

walking facility (in m). (4) T is the time horizon (in s). (5) qc(x, y, t) expresses the number

of pedestrians who cross a unit width of Γco and describes the time-varying demand of Group

c (in ped/m/s). (6) ρc(x, y, t) is a time-varying function that denotes the pedestrian density

of Group c (in ped/m2). (7) Fc(x, y, t) := (f c1(x, y, t), f c

2 (x, y, t)) represents the flow vector for

Group c, where f c1(x, y, t) is the flow flux in the x-direction and f c

2(x, y, t) is the flow flux in

the y-direction (in ped/m/s). By default, Fc(x, y, t) · n = 0, where n denotes the unit outward

normal of Γch. (8) vc(x, y, t) is the local walking speed of Group c (in m/s). (9) τc(x, y, t) is

the local cost of Group c (in s). (10) Φc(x, y, t) is the instantaneous walking time potential of

Group c from location (x, y) to Γcd, at which Φc = 0 (in s).

Finally, Γ = Γco

⋃Γc

d

⋃Γc

h, c ∈ {a, b}, (x, y) ∈ Ω and t ∈ T .

2.1 Flow conservation equation

Similar to most physical systems, the density, velocity and flow of pedestrians follow the

physical principle of conservation. The conservation of mass for bi-directional pedestrian flows

is given by

ρct(x, y, t) + ∇ · Fc(x, y, t) = 0, (2.1)

where ρct(x, y, t) = ∂ρc(x,y,t)

∂tand ∇ · Fc(x, y, t) =

∂fc

1(x,y,t)∂x

+∂fc

2(x,y,t)∂y

.

2.2 Walking speed function

The local walking speeds of the two groups depend on the densities and traveling directions

of the pedestrians in the walking facility and are determined by (see [38])

va(ρ, Ψ) = vf exp(− α(ρa + ρb)2

)exp

(− β(1 − cos(Ψ))(ρb)2

), (2.2)

vb(ρ, Ψ) = vf exp(− α(ρa + ρb)2

)exp

(− β(1 − cos(Ψ))(ρa)2

), (2.3)

where ρ := {ρa(x, y, t), ρb(x, y, t)}. Here, vf is the free-flow walking speed of pedestrians, α,

β, are the model parameters, and Ψ := Ψ(x, y, t) is the intersecting angle between the two

1544 ACTA MATHEMATICA SCIENTIA Vol.29 Ser.B

pedestrian streams of Group a and Group b, respectively, at location (x, y), as shown in Figure

1.

Fig.1 Angle Ψ between Streams a and b

For the case in which there is only single stream a, i.e., ρb = 0, (2.2) and (2.3) can be

formulated as

va(ρ, Ψ) = vf exp(− α(ρa)2

),

vb(ρ, Ψ) = vf exp(− α(ρa)2

)exp

(− β(1 − cosΨ)(ρa)2

).

However, vb(ρ, Ψ) in this situation is, effectively, only a direction-dependent potential speed if

there is potentially a single pedestrian trying to walk in direction Ψ against Stream a. Therefore,

the actual speed of Stream b is not relevant. Also, the speed is identical to va(ρ, Ψ) if this

potential pedestrian happens to walk parallel to Stream a (i.e., Ψ = 0).

2.3 Local cost function

The local cost, τc, as a function of location and time is assumed to depend on walking

speed alone and can be specified as

τc(x, y, t) : = τ(vc(ρ, Ψ)) =1

vc(ρ, Ψ), (2.4)

which describes the walking time per unit distance of movement incurred by a pedestrian in

Group c in the walking facility.

2.4 Definitional relationship

For each Group c ∈ {a, b}, along the direction of the flow vector (or movement), flow

intensity, which is determined as the norm of the pedestrian flow, is equal to the product of

speed and density,

‖ Fc(x, y, t) ‖= vc(ρ, Ψ)ρc(x, y, t), (2.5)

where

‖ Fc(x, y, t) ‖=(f c1 (x, y, t)2 + f c

2(x, y, t)2) 1

2 .

2.5 Path choice constraints

It is assumed that a pedestrian in each group at location (x, y) ∈ Ω will choose a path

that minimizes his or her travel cost to the destination, based on the instantaneous travel cost

information that is available at the time of making a decision. Then, we have

τc(x, y, t)Fc(x, y, t)

‖ Fc(x, y, t) ‖+ ∇Φc(x, y, t) = 0, (2.6)

where

∇Φc =(∂Φc

∂x,∂Φc

∂y

).

No.6 Y.Q. Jiang et al: A REACTIVE DYNAMIC CONTINUUM USER EQUILIBRIUM MODEL 1545

Based on (2.4)–(2.6),

Fc(x, y, t) = −vc(ρ, Ψ)ρc(x, y, t)

τc(x, y, t)∇Φc(x, y, t)

= −ρc(x, y, t)(vc(ρ, Ψ))2∇Φc(x, y, t), (2.7)

and

‖ ∇Φc(x, y, t) ‖= τc(x, y, t). (2.8)

From (2.7), assuming that ρc(x, y, t) �= 0, the following property is obtained [19, 39–43].

Fc(x, y, t)//(−∇Φc(x, y, t)), (2.9)

where “//” means that the two vectors, Fc(x, y, t) and ∇Φc(x, y, t), are parallel and pointing

in the opposite direction. Furthermore, (2.8) can be shown as follows.

For any used path p, from location (x, y) ∈ Ω to destination (x0, y0) ∈ Γcd, if we integrate

the local travel cost along the path, then the total cost incurred by pedestrians can be obtained

by

τcp(x, y, t) =

∫p

τc(x, y, t)ds =

∫p

τc(x, y, t)Fc(x, y, t)

‖ Fc(x, y, t) ‖· ds = −

∫p

∇Φc(x, y, t) · ds

= −(Φc(x0, y0, t) − Φc(x, y, t)) = Φc(x, y, t), (2.10)

using (2.6)–(2.9), the fact that Fc(x,y,t)

‖Fc(x,y,t)‖ is a unit vector that is parallel to ds along the path,

and Φc(x0, y0, t) = 0, ∀(x0, y0) ∈ Γcd, t ∈ T. Hence, the total instantaneous travel cost at time

t ∈ T is independent of the used paths.

In contrast, for an unused path p, between location (x, y) ∈ Ω and destination (x0, y0) ∈ Γcd,

the total walking time incurred by the pedestrian is

τcp(x, y, t) =

∫p

τc(x, y, t)ds ≥

∫p

τc(x, y, t)Fc(x, y, t)

‖ Fc(x, y, t) ‖· ds = −

∫p

∇Φc(x, y, t) · ds

= −(Φc(x0, y0, t) − Φc(x, y, t)) = Φc(x, y, t), (2.11)

using (2.6)–(2.9).

The inequality in the foregoing derivation occurs because of some segments along path

p whose normal vectors Fc(x,y,t)

‖Fc(x,y,t)‖ are not parallel, ds; thus, ds > ( Fc(x,y,t)

‖Fc(x,y,t)‖) · ds for those

segments. Hence, for any unused paths, the total instantaneous travel cost is greater than or

equal to that of the used paths. In this way, the model guarantees that pedestrians choose their

paths in the walking facility in a user-optimal manner with respect to the instantaneous travel

information [44], that is, the pedestrian flow conditions at the time.

The model can now be formulated as the following set of differential equations.⎧⎪⎪⎨⎪⎪⎩

ρct(x, y, t) + ∇ ·Fc(x, y, t) = 0,

Fc(x, y, t) = −ρc(x, y, t)(vc(ρ, Ψ))2∇Φc(x, y, t),

‖ ∇Φc(x, y, t) ‖= τc(x, y, t),

(2.12)

where τc(x, y, t) are determined by (2.2)–(2.4), respectively. Note that angle Ψ in (2.2) and

(2.3) can be obtained by two potential functions due to (2.7). Then, we have

cos(Ψ) =∇Φa(x, y, t)

‖ ∇Φa(x, y, t) ‖·

∇Φb(x, y, t)

‖ ∇Φb(x, y, t) ‖.

1546 ACTA MATHEMATICA SCIENTIA Vol.29 Ser.B

The system (2.12) is subject to the following initial boundary conditions.

Fc(x, y, t) · n(x, y) = qc(x, y, t), ∀(x, y) ∈ Γco, (2.13)

ρc(x, y, 0) = ρc0(x, y), ∀(x, y) ∈ Ω, (2.14)

Φc(x, y, t) = 0, ∀(x, y) ∈ Γcd, (2.15)

where n(x, y) is a unit normal vector that points to the domain boundary, and ρc0(x, y) is the

initial pedestrian density for each group.

3 Solution Algorithms

Here, the cell-centered FVM for an approximation of the 2D conservation law equation and

the FMM for the Eikonal-type problem on unstructured meshes, respectively, are introduced

to solve the system (2.12).

3.1 Finite Volume Spatial Discretization

Let T h :=⋃

Ti∈T h

Ti be an acute triangulation of computational domain Ω with Np nodes,

Nt cells and Nf faces. The size of the discretization is defined by h := superTi∈T h{hTi}, where

hTiis the exterior diameter of triangular cell Ti. We label the neighboring triangles of Ti as

Tik, with a counter-clockwise convention as sketched in Figure 2.

Fig.2 A typical unstructured triangular grid

A cell-centered FVM is considered, i.e., each cell represents a control volume (CV). The

unknowns are stored in the geometric center of the cells. Integrating (2.1) over Ti and applying

the Gauss’s theorem yields∫Ti

∂ρc

∂tdxdy +

∮∂Ti

Fc · nds = 0, ∀ Ti ∈ T h, (3.16)

where ∂Ti denotes three edges of Ti, and n is the outer unit normal pointing to ∂Ti. Assuming

that ρci is the average quantity stored in the cell center of Ti, we obtain the following formula.

∂ρci

∂t+

1

Ai

∮∂Ti

Fc · nds = 0, ∀ Ti ∈ T h, (3.17)

where Ai refers to the area of Ti. To obtain the semi-discrete form of (3.17), the boundary inte-

gral term needs to be evaluated. Let lik denote the k-th edge of Ti and nik be the corresponding

normal vector, k = 1, 2, 3. The integral expression is discretized by summing the flux vectors

over each edge of the triangular cell. Utilizing a mid-point quadrature formula generates

∮∂Ti

Fc · nds =

3∑k=1

(Fc

ik · nik

)mik

|lik|, ∀ Ti ∈ T h, (3.18)

No.6 Y.Q. Jiang et al: A REACTIVE DYNAMIC CONTINUUM USER EQUILIBRIUM MODEL 1547

where Fcik is the numerical flux through the middle point mik of lik.

Fig.3 Schematic diagram of two adjacent triangular cells

To obtain the upwind mechanism, high-order accuracy and essential stability of the nu-

merical scheme, a splitting into the normal flux function Fc ·n at location Xjk= (xjk

, yjk), k =

1, 2, 3, 4 (see Figure 3) is adopted. For instance, Lax-Friedrichs splitting [45] is used.

(Fc · n)± =1

2[(F · n)(ρc) ± λρc], (3.19)

where λ := vc(ρ, Ψ) is the upper bound of the spectrum of the Jacobi of the normal flow

function. Here, Xjk, k = 2, 3 is the barycentric coordinate of the triangles, and Xjk

, k = 1, 4 is

the vertex of the triangles. The unknown concentration at a vertex in the mesh is obtained via

the inverse distance weighting interpolation method [46].

The normal numerical flux across the cell boundary l between two neighboring cells can

be written as

(Fc · n)m = (Fc · n)L + (Fc · n)R. (3.20)

For a first-order (1st) scheme, let (Fc · n)L = [F · n(ρcj2

)]+ and (Fc · n)R = [F · n(ρcj3

)]− by

assuming constant variables in each cell. Here it is postulated that the variables are the adjacent

piecewise constant values in each cell. Applying a MUSCL reconstruction technique yields a

second-order (2nd) spatial accuracy scheme [47]. The detailed scheme is

(Fc · n)L = G1((Fc · n)+j1 , (F

c · n)+j2 , (Fc · n)+j3 ), (3.21)

(Fc · n)R = G2((Fc · n)−j2 , (F

c · n)−j3 , (Fc · n)−j4), (3.22)

with

G1(q1, q2, q3) = q2 +1

4[(1 − ε)Δ− + (1 + ε)Δ+]q2, (3.23)

G2(q2, q3, q4) = q3 −1

4[(1 − ε)Δ+ + (1 + ε)Δ−]q3, (3.24)

where (Fc ·n)±jk= [(F ·n)(ρc

jk)]±, k = 1, 2, 3, 4, and Δ+, Δ− indicate the forward and backward

difference operators, respectively, e.g., Δ+qi = qi+1 − qi, Δ−qi = qi − qi−1, i = 2, 3. The value

of ε is 13 in this paper. Note that for boundary CV, the values of ρc at Xj3 , Xj4 are obtained by

extrapolation from the inside computational domain. To avoid great change in the mesh size

of neighboring cells, Δ+q2 in (3.23) and Δ−q3 in (3.24) are modified as follows.

Δ+q2 =2l1

l1 + l2(q3 − q2), Δ−q3 =

2l2l1 + l2

(q3 − q2),

1548 ACTA MATHEMATICA SCIENTIA Vol.29 Ser.B

where li, i = 1, 2 is the length from the centroid of the left or right cell to the mid-point of the

edge l.

Furthermore, a slope limiter functions L(θ−, θ+) can be imposed on (3.23) and (3.24) to

suppress spurious oscillations in the process of computation. The two formulas are rewritten as

G1(q1, q2, q3) = q2 +L(θ−, θ+)

4[(1 − L(θ−, θ+)ε)Δ− + (1 + L(θ−, θ+)ε)Δ+]q2, (3.25)

G2(q2, q3, q4) = q3 −L(θ−, θ+)

4[(1 − L(θ−, θ+)ε)Δ+ + (1 + L(θ−, θ+)ε)Δ−]q3. (3.26)

A double-parameter minmod limiter [48] is adopted due to its robustness.

L(θ−, θ+) = minmod(1, θ−) + minmod(1, θ+) − 1.

where θ− = q2−q1

q3−q2, θ+ = q4−q3

q3−q2.

3.2 Calculation of ∇Φc on CV surfaces

In (2.7), the values of ∇Φc at each CV boundary are unknown. Assuming that Φc is

piece-wise linear on each triangle, it can be written as

Φc(x, y)|Ti=

3∑k=1

NkΦck, ∀Ti ∈ T h, (3.27)

where Nk, k = 1, 2, 3 is a Lagrangian interpolation basis function on the triangular meshes and

Φck is the value of Φc at the k-th vertex of triangle Ti. Then, ∇Φc is piece-wise constant on

each triangle, and

∇Φc(x, y)|Ti=

3∑k=1

∇NkΦck, ∀Ti ∈ T h. (3.28)

The approximate value of Φc at each vertex of mesh is calculated by solving the nonlinear

Eikonal equation. Several numerical methods can be used for this purpose, such as the group

marching method (GMM), the FMM or the fast sweeping method (FSM), among others. The

FMM is chosen, as its yields are consistent, accurate and highly efficient, based on entropy-

satisfying upwind schemes and fast sorting techniques.

Here a brief review of the upwind finite difference discretization for the Eikonal equation

is given. Consider simplex X0X1X2 in the triangular meshes. We assume that Φc at X0

is to be computed and that Φcl ,∇Φc

l , l = 1, 2 at X1 and X2 are given or have already been

computed. Define unit vectors Nl = X0−Xl

|X0−Xl|, l = 1, 2 and a 2×2 nonsingular matrix N =

(N1

N2

)

(X0, X1 and X2 are not lined up). The directional derivatives in the N1 and N2 directions are

approximated by

DlΦc =

Φc0 − Φc

l

|X0 − Xl|, l = 1, 2, 1st formula, (3.29)

DlΦc = 2

Φc0 − Φc

l

|X0 − Xl|− Nl · (∇Φc

l )′, l = 1, 2, 2nd formula, (3.30)

where the superscript “′” stands for vector or matrix transposition. The approximation direc-

tional derivatives are linked to the gradient by

DΦc = N · (∇Φc)′ + O(hα), (3.31)

No.6 Y.Q. Jiang et al: A REACTIVE DYNAMIC CONTINUUM USER EQUILIBRIUM MODEL 1549

where D =

[D1

D2

], h = max{|X0 − X1|, |X0 − X2|}, and α = 1 or 2, depending on whether

(3.29) or (3.30) is used. Substituting ∇Φc, which was obtained by (3.31), into (2.8), we obtain

the (quadratic) equation that defines the unknown Φc0:

(DΦc)′(NN ′)−1(DΦc) = (τc0 )2. (3.32)

Φc0 can be updated if the upwind criterion is satisfied: the characteristic direction should point

to simplex X0X1X2 and is equivalent to

(NN ′)−1DΦc > 0. (3.33)

Then, the value of Φc0 at X0 is determined by

Φc0 = min

⎧⎨⎩

solution of (3.29) or (3.30) , if (3.33) holds,

minl=1,2

{Φcl + |X0 − Xl|τ

c0}, others.

(3.34)

With the FMM, the values of Φc at all mesh nodes are considered in an order that is consistent

with the way fronts propagate. This leads us to a single-pass algorithm, which is the advantage

of this method in solving front propagation problems with a lower degree of computational

complexity. Following Ref. [36], the procedure of the algorithm is given as follows. First, the

mesh points are divided into three sets, i.e., Accepted, Considered and Far. For example, tag

the points on Γco as Accepted (Φc = 0 on Γc

d) and then tag all points adjacent to those in

Accepted set as Considered. Finally, tag all other points as Far. The fast-marching algorithm

is briefly described as follows.

• The point with the lowest value in the Considered set is removed from it and then added

to the Accepted set; its unaccepted neighbors are added to the Considered set.

• The values of all points in the Considered set are recomputed using (3.34).

• Repeat the process until all of the points are in the Accepted set.

For more detailed depictions of the algorithm, refer to Refs. [36] and [37].

3.3 Time discretization

To obtain 2nd accuracy in time, the explicit TVD Runge-Kutta time-stepping method [49]

is chosen to discretize (3.17) as follows.

(ρc)(1)i = (ρc)n

i −Δt

Ai

3∑k=1

(Fc · n)nmk

|lik|, (3.35)

(ρc)n+1i =

1

2

[(ρc)n

i + (ρc)(1)i −

Δt

Ai

3∑k=1

(Fc · n)(1)mk|lik|

]. (3.36)

Here, time step Δt needs to satisfy the Courant-Friedrichs-Lewy (CFL) condition. For a 2D

MUSCL scheme on unstructured grids, the stability condition is

Δt

mini Ai

maxρc

∣∣∣d(Fc · n)

dρc

∣∣∣ · maxi,k

|lik| ≤ 1,

where i = 1, 2, · · · , Nt, k = 1, 2, 3.

1550 ACTA MATHEMATICA SCIENTIA Vol.29 Ser.B

4 Numerical Example

In this section, we consider a railway platform that is 100m by 50m in size, as shown in

Figure 4. Group c enters the platform from segment Γco with a width of 20m and leaves the

platform from segment Γcd with a width of 20m.

Fig.4 Overview of the railway platform

The time step is set to be 240s. Initially, the platform is empty. The inlet flow rate,

qc(x, y, t), on Γco represents time-varying demand and is given as follows.

qa(0, y, t) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

t

30× ma, t ∈ [0, 30],

ma, t ∈ [30, 90],

−ma ×t − 120

30, t ∈ [90, 120],

0, t ∈ [120, 300],

with ∀y ∈ [15m, 35m], and

qb(x, 0, t) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

t

30× mb, t ∈ [0, 30],

mb, t ∈ [30, 90],

−mb ×t − 120

30, t ∈ [90, 120],

0, t ∈ [120, 300],

with ∀x ∈ [65m, 85m], where ma and mb denote, respectively, the peaks of the inlet flow rate

at the two entrances and we take ma = 1.0ped/m/s and mb = 0.5ped/m/s for the calculation.

Then, flow vector Fa = (qa, 0) on Γao and Fb = (0, qb) on Γb

o. The free-flow walking speed is

taken as 1.034m/s and two model parameters, α, β in (2.2) and (2.3), are taken as 0.075, 0.019,

respectively (see [38]).

The validation for the convergence of numerical solutions generally involves comparing

them with the exact solutions of the problem. However, the exact solutions of the system

model (2.12) are unknown. We thus have to use other methods (e.g., by refining the mesh) to

convince ourselves about the convergence of the numerical solutions.

The numerical experiments on four different sizes of unstructured meshes are tested: Mesh

1 with h = 1.6m; Mesh 2 with h = 0.8m; Mesh 3 with h = 0.4m; and Mesh 4 with h = 0.2m.

No.6 Y.Q. Jiang et al: A REACTIVE DYNAMIC CONTINUUM USER EQUILIBRIUM MODEL 1551

Comparing the numerical solutions obtained using the numerical methods mentioned in Ref.

[19], i.e., the 1st order accuracy upwind scheme for the conservation law equation coupled with

the 1st order FSM on orthogonal grids, indicates that the two numerical methods achieve almost

the same results on different types of grids (see Figures 5 and 6). Here, four different partition

sizes that correspond to the aforementioned mesh sizes are applied to generate orthogonal grids.

The density curves of Group a along the center line, x = 50m, of the computational domain

with two different methods are shown in Figure 5. Figure 6 depicts the density curves of Group

b along the center line, y = 25m, of the computational domain with two different methods.

From these figures, it can be seen that the densities as functions of time in decreasing discrete

sizes converge toward a unique solution. In addition, it is also believed that the numerical

solutions are stable and can be found with the finest mesh.

Y

ρ

10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 100*50200*100300*150400*200

Y

ρ

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1 Mesh 1Mesh 2Mesh 3Mesh 4

Fig.5 (Color online) Density plot of Group a at x = 50m and at t = 90s

X

ρ

20 40 60 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 100*50200*100300*150400*200

X

ρ

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1 Mesh 1Mesh 2Mesh 3Mesh 4

(a) upwind (b) MUSCL

Fig.6 (Color online) Density plot of Group b at y = 25m and t = 120s

The densities of the bi-directional pedestrian flows in the railway platform are plotted in

Figure 7 at different times. This figure clearly shows that the walking area is not fully occupied

based on the optimal strategy for making a path-choice decision. It can also be seen that the

flows spread over the center of the groups and form two strips that are composed of pedestrians

1552 ACTA MATHEMATICA SCIENTIA Vol.29 Ser.B

walking in the same direction; then, each pedestrian in his or her group attempts to move

toward the center of the group. The two streams of pedestrians walking in crossing directions

to their different destinations always tend to separate. The dynamic continuum model of bi-

directional pedestrian flows can reproduce self-organized phenomena [8, 12, 20–22], as dynamic

lane formation can be also observed in such flows.

X

Y

0 20 40 60 80 1000 0

20 20

40 40

0.2 0.4 0.6 0.8 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4

X

Y

0 20 40 60 80 1000 0

20 20

40 40

0.2 0.4 0.6 0.8 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4

(a) t = 60s (b) t = 60s

X

Y

0 20 40 60 80 1000

20

40

0.2 0.4 0.6 0.8 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4

X

Y

0 20 40 60 80 1000

20

40

0.2 0.4 0.6 0.8 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4

(c) t = 90s (d) t = 90s

X

Y

0 20 40 60 80 1000

20

40

0.2 0.4 0.6 0.8 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4

X

Y

0 20 40 60 80 1000

20

40

0.2 0.4 0.6 0.8 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4

(e) t = 120s (f) t = 120s

X

Y

0 20 40 60 80 1000

20

40

0.2 0.4 0.6 0.8 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4

X

Y

0 20 40 60 80 1000

20

40

0.2 0.4 0.6 0.8 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4

(g) t = 180s (h) t = 180s

Fig.7 (Color online) Density plot (left: Group a; right: Group b)

Figure 8 depicts the flow vector (2.7) of each group at different times, which helps us to

observe the distribution and evolution of the flows. Strip formation in the two intersecting

pedestrian streams is displayed. At the end of the modeling period, i.e., t = 240s, almost all

pedestrians get away from the railway platform.

No.6 Y.Q. Jiang et al: A REACTIVE DYNAMIC CONTINUUM USER EQUILIBRIUM MODEL 1553

At the beginning of the modeling period when the two groups do not interact, the pedes-

trians move forward toward their goal, with almost none of them changing direction. As time

marches on, the two groups arrive in the common area and are then interactive and passed by

each other. As a result, the primary walking directions of the two groups vary. In addition,

high-density values can be seen near the two exits, as the crowded conditions force the pedes-

trians’ speed to decrease as they line up to walk through the relatively narrow exits. When

pedestrians want to pass through a bottleneck (e.g., coming into being in the common area and

at the exits), interaction and negotiation become important to crossing flows and crowd flows

[22].

X

Y

0 20 40 60 80 1000

10

20

30

40

50

X

Y

0 20 40 60 80 1000

10

20

30

40

50

(a) t = 30s (b) t = 60s

X

Y

0 20 40 60 80 1000

10

20

30

40

50

X

Y

0 20 40 60 80 1000

10

20

30

40

50

(c) t = 90s (d) t = 120s

X

Y

0 20 40 60 80 1000

10

20

30

40

50

X

Y

0 20 40 60 80 1000

10

20

30

40

50

(e) t = 150s (f) t = 180s

X

Y

0 20 40 60 80 1000

10

20

30

40

50

X

Y

0 20 40 60 80 1000

10

20

30

40

50

(g) t = 210s (h) t = 240s

Fig.8 (Color online) Flow vector plot (red: Group a; green: Group b)

1554 ACTA MATHEMATICA SCIENTIA Vol.29 Ser.B

5 Conclusion

In this work, the reactive dynamic user equilibrium model reported by Hughes [16] and

Huang et al. [19] is extended to simulate two groups of pedestrians walking in crossing direc-

tions in a continuous walking facility. In this model, strategic path choice is the decision each

pedestrian in a group makes to minimize the travel cost to his or her destination in a reactive

manner based on the instantaneous available information. By numerically solving the model,

a number of results, e.g., the formation of strips composed of pedestrians walking to the same

destination, are observed. Lane or strip formation embodies pedestrians’ cooperative strategy

when moving in different directions [22]. A relatively high degree of density exists at the bot-

tlenecks, i.e., the two exits, which causes traffic congestion. The numerical results obtained

with this continuum model can serve as a fundamental basis for congestion control and traffic

management.

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