An efficient discontinuous Galerkin method on triangular meshes for a

pedestrian flow model

Yinhua Xia1, S.C. Wong2, Mengping Zhang3, Chi-Wang Shu4 and William H.K. Lam5

Abstract

In this paper, we develop a discontinuous Galerkin method on triangular meshes to solve

the reactive dynamic user equilibrium model for pedestrian flows. The pedestrian density in

this model is governed by the conservation law in which the flow flux is implicitly dependent

on the density through the Eikonal equation. To solve the Eikonal equation efficiently at

each time level, we use the fast sweeping method. Two numerical examples are then used to

demonstrate the effectiveness of the algorithm.

Key Words: Pedestrian flow; continuum modeling; reactive dynamic user equilibrium;

discontinuous Galerkin method; fast sweeping method; triangular mesh; WENO scheme

1Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China.E-mail: xiayh@mail.ustc.edu.cn.

2Department of Civil Engineering, The University of Hong Kong, Hong Kong, China. E-mail:hhecwsc@hkucc.hku.hk.

3Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China.E-mail: mpzhang@ustc.edu.cn.

4Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail:shu@dam.brown.edu.

5Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong,China. E-mail: cehklam@polyu.edu.hk.

1

1 Introduction

A systematic framework for the dynamic macroscopic modeling of pedestrian flow problems

was given in [1], but the user equilibrium concept was not explicitly considered. To address

this issue, Hoogendoorn and his colleagues developed a predictive user equilibrium model,

in which pedestrians were assumed to have perfect information to make their route choice

decisions over time [2, 3, 4], to model the dynamic route choice behavior of pedestrians.

This modeling approach is useful in representing a more strategic level of route choice de-

cisions whereby pedestrians accumulate their travel information from their daily experience

or from other sources. However, in some applications, pedestrians may not have predictive

information when they make a route choice decision [5], and, very often, they may have

to rely on instantaneous information and change their choice in a reactive manner when

walking through a facility (see [6] for the difference between predictive and reactive dynamic

user equilibrium principles). Recently, Huang et al. [7] developed a dynamic macroscopic

model of pedestrian flow in the context of the reactive user equilibrium principle, in which

pedestrians from a given location choose the path that minimizes their individual walking

cost to a destination based on the instantaneous travel cost information that is available to

them at the time of decision-making.

In [7], pedestrian density is governed by the scalar two-dimensional conservation law,

and the flow flux is implicitly dependent on the density through the Eikonal equation. The

weighted essentially non-oscillatory (WENO) scheme on rectangular mesh has been devel-

oped to solve the model. To enhance the applicability of this pedestrian model, we develop

an efficient numerical method to solve the pedestrian flow model on an arbitrary domain,

in which we use the discontinuous Galerkin (DG) method on triangular mesh to solve the

conservation law, coupled with the fast sweeping method to solve the Eikonal equation.

The DG method is in the class of finite element methods and uses discontinuous, piece-

wise polynomials as the solution and the test space. The first DG method was introduced

in 1973 by Reed and Hill [8] in the framework of a neutron transport problem, i.e., a time-

2

independent linear hyperbolic equation. It was first developed for hyperbolic conservation

laws that contained first derivatives by Cockburn et al. in a series of papers [9, 10, 11, 12] in

which they established a framework to solve nonlinear time-dependent problems easily. This

framework uses explicit, nonlinearly stable, high-order Runge-Kutta time discretizations [13]

and DG discretization in space with exact or approximate Riemann solvers as the interface

fluxes and total variation bounded (TVB) nonlinear limiters to achieve non-oscillatory prop-

erties for strong shocks. The DG method has rapidly found applications in such diverse fields

as aeroacoustics, electro-magnetism, gas dynamics, granular flows, magneto-hydrodynamics,

meteorology, the modeling of shallow water, oceanography, oil recovery simulation, semicon-

ductor device simulation, the transport of contaminants in porous media, turbomachinery,

turbulent flows, viscoelastic flows, and weather forecasting, among many others. For a de-

tailed description of the method and for its implementation and applications for conservation

laws, we refer interested readers to [14, 15, 16, 17, 18, 19, 20, 21, 22].

Unlike the usual conservation law, in the pedestrian flow model discussed in this paper,

the flow flux is implicitly dependent on pedestrian density through an Eikonal equation,

which is a special steady state Hamilton-Jacobi equation. There has been much algorithm

development and many simulations of the Eikonal equation. The fast marching method

[23, 24, 25] and the fast sweeping method [26, 27, 28, 29] are designed to solve a nonlinear

discretized system directly and efficiently by exploiting the causality of the Eikonal equations.

The fast marching method has the complexity of O(N log N), where N is the total number

of the mesh points, whereas the fast sweeping method has the complexity of O(N). In [30],

these two methods are compared for various numerical examples. For a particular problem

on a fixed grid, one method may be faster than the other. When the grid is more refined,

however, the fast sweeping method will eventually be faster. This is the reason we choose

the fast sweeping method in our algorithm.

The organization of the paper is as follows. In Section 2, we give a brief description

of the model for pedestrian flows. Section 3 is devoted to a concise description of the

3

numerical algorithm, including the discontinuous Galerkin (DG) spatial discretization for the

conservation law, the fast sweeping method for the Eikonal equation, and the total variation

diminishing (TVD) Runge-Kutta temporal discretization. Numerical examples for testing

the performance of the numerical algorithm for the pedestrian flow model are presented in

Section 4. Finally, we give concluding remarks in Section 5.

2 The pedestrian flow model

We consider the two-dimensional domain Ω, with the inflow boundary Γi, the outflow bound-

ary Γo, and the solid wall boundary Γw (∂Ω = Γi∪Γo∪Γw). The pedestrian flows are governed

by the following conservation law.{ρt(x, y, t) + ∇ · f (ρ(x, y, t)) = 0, ∀(x, y) ∈ Ω,

ρ(x, y, 0) = ρ0(x, y), ∀(x, y) ∈ Ω,(1)

where ρ(x, y, t) is the pedestrian density at location (x, y) at time t, ρ0(x, y) is the initial

density, f (x, y, t) = (f1(x, y, t), f2(x, y, t)) is the pedestrian flow vector with fluxes f1(x, y, t)

and f2(x, y, t) in the x and y directions, respectively. We define a cost potential function

φ(x, y, t) that satisfies the Eikonal equation:{‖∇φ(x, y, t)‖ = c(x, y, t), ∀(x, y) ∈ Ω,

φ(x, y, t) = 0, ∀(x, y) ∈ Γo,(2)

where c(x, y, t) is the local travel cost that is related to the pedestrian density ρ(x, y, t) and

isotropic walking speed u(x, y, t) by the following two equations:

u(x, y, t) = U(ρ, x, y, t) := umax(x, y)

(1 − ρ(x, y, t)

ρmax(x, y)

), (3)

c(x, y, t) = C(u, x, y, t) :=1

u(x, y, t), (4)

where umax(x, y) and ρmax(x, y) are the free-flow speed and jam density, respectively. By

default, ‖f‖ = (f 21 (x, y, t) + f 2

2 (x, y, t))1/2

= u(x, y, t)ρ(x, y, t).

To ensure the reactive user equilibrium condition in which pedestrians choose a path

that minimizes their total cost to a destination according to instantaneous pedestrian flow

information, we have f (x, y, t)//−∇φ(x, y, t) (see [7]).

4

3 The numerical procedure

Equation (1) is a scalar two-dimensional hyperbolic conservation law. There has been much

algorithm development of this type of equations when the flux f is given explicitly. In this

paper, we choose the discontinuous Galerkin (DG) method to solve the conservation law.

This method is well-suited to complex geometries as they can be applied on unstructured

grids. In addition, the DG method can also handle non-conforming elements, where the

grids are allowed to have hanging nodes, and is highly parallelizable, as the elements are

discontinuous and the inter-communications are minimal.

In our pedestrian flow model, the flux f is given implicitly. We need to solve the Eikonal

equation (2) at each time level to compute the flux f . We choose the fast sweeping method

on triangular meshes to solve the Eikonal equation (2), which is an efficient iterative method

for stationary Hamilton-Jacobi equations. The computational complexity of the algorithm

is nearly optimal with O(N log N) for sorting at the initial step and O(N) for the iterative

steps, where N is the total number of grid points.

Starting from density ρn at time level tn, to obtain density ρn+1 at time level tn+1 under

the Euler forward time discretization, we need to compute the following two steps.

1. Solve the Eikonal equation (2) by the fast sweeping method (to be discussed in Section

3.1),

2. Solve the conservation law (1) by the discontinuous Galerkin method (to be discussed

in Section 3.2).

3.1 The fast sweeping algorithm

In this section, we illustrate the fast sweeping algorithm to solve the Eikonal equation

(2). Consider a node xj on a triangular mesh at which the solution φj is to be com-

puted and two neighboring nodes, xj1 and xj2, at which the values of φj1 and φj2 and

their derivatives (∂xφjl, ∂yφjl, l = 1, 2) are known or have been computed. The direc-

5

tional derivative in the directions defined by the linearly independent row unit vectors

Pl = (xj − xjl)/ ‖xj − xjl‖ , l = 1, 2 is approximated by

Dlφ =φj − φjl

|xj − xjl|, l = 1, 2 (first order formula), (5)

Dlφ = 2φj − φjl

|xj − xjl|− Pl · [∂xφjl, ∂yφjl], l = 1, 2 (second order formula). (6)

We have

Dφj ≈ P∇φj, P =

[P1

P2

], (7)

where Dφj = [D1φj, D2φj]T . Combine this equation with the Eikonal equation (2), and we

can now write an equation for φj:

(Dφj)T (PP T )−1Dφj = (c(xj, yj, t))

2. (8)

Suppose we are computing the value of φj from some triangle, with xj being one of its

vertices. The computed solution φj can be accepted only if the computed approximate −∇φj

lies inside the simplex. This restriction, which is called upwind criteria, is equivalent to two

inequalities: D1φj ≥ (P1 · P T2 )D2φj and D2φj ≥ (P1 · P T

2 )D1φj.

We define the local solver in a triangle, given φj1 and φj2, to obtain φj.

Local solver:

1. Solve the equation (8) to obtain φj .

2. If (D1φj ≥ (P1 · P T2 )D2φj and D2φj ≥ (P1 · P T

2 )D1φj), then

φj = min(φj, φj)

else

φj = min(φj, |xj − xj1| c(xj , yj, t), |xj − xj2| c(xj , yj, t))

endif.

6

The fast sweeping method is a Gauss-Seidel type update process. On a rectangular

mesh, four sweeping directions can be considered. We would have lower left to upper right

(i = 1 : I, j = 1 : J); upper left to lower right (i = 1 : I, j = J : 1); lower right to upper

left (i = I : 1, j = 1 : J); and upper right to lower left (i = I : 1, j = J : 1). However,

on a triangular mesh, we need to give systematic ordering that can cover all directions of

information propagation. Following [29], we set at least three noncollinear points as the

reference points, then sweep through all of the nodes according to ‖x − xref‖p (p = 1 or 2)

in both the ascending and descending order.

The fast sweeping algorithm on a triangular mesh

1. Initialization:

(a) Sort all nodes according to the lp distance to the chosen reference points xiref , i =

1, · · · , R.

(b) Assign the exact values φj = 0 and Dφj on the boundary Γo and keep these values

fixed during the iterations. At all other vertices, assign the larger positive values

φj = ∞, and these values will be updated in later iterations.

2. Gauss-Seidel iteration:

(a) Choose the direction of sweep and a corresponding ordering.

(b) Loop through all of the nodes in the chosen order and apply the local solver.

(c) Repeat for the other directions of sweep.

(d) Repeat until convergence.

We note that the local solver is quite useful for the acute triangulations used in the

following numerical tests. It can lead to the numerical instability of the scheme when used

for arbitrary unstructured meshes, because the causality relationship requires φj to depend

on the smaller values of φ at the grid points adjacent to xj. One possible solution was

suggested in [31].

7

3.2 The discontinuous Galerkin methods

In this section, we show how to discretize equation (1) in space by the DG method. Once a

triangulation Th of Ω has been obtained, we define the finite element space V kh as

V kh ≡

{v ∈ L2(Ω) : v|K ∈ P k(K), ∀K ∈ Th

}, (9)

where K is the element of Th and P k(K) denotes the space of the polynomial in K of the

degree at most k. Note that the functions in vh are allowed to have discontinuities across

the element interfaces.

First, we multiply (1) by vh in V kh , integrate over the element K of triangulation Th, and

replace the exact solution ρ by its approximation ρh ∈ V kh :

d

dt

∫K

ρh(x, y, t)vh(x, y)dxdy +

∫K

∇ · f(ρ(x, y, t))vh(x, y)dxdy = 0, ∀vh ∈ V kh . (10)

Integrating by parts we formally obtain

d

dt

∫K

ρh(x, y, t)vh(x, y)dxdy +∑e∈∂K

∫e

f (ρ(x, y, t)) · ne,Kvh(x, y)ds

−∫

K

f (x, y, t) · ∇vh(x, y)dxdy = 0, ∀vh ∈ V kh ,

where ne,k is the outward unit normal to edge e. Notice that f · ne,K does not have a

precise meaning, because ρh is discontinuous at (x, y) ∈ e ∈ ∂K. We replace f · ne,K with

he,K(ρint(K), ρext(K)), which is any consistent two-point monotone Lipschitz flux, consistent

with f · ne,K. Here we choose the Lax-Friedrichs flux:

he,K(a, b) =1

2[f (a) · ne,K + f(b) · ne,K − α(b − a)],

α = maxmin(a,b)≤ρ≤max(a,b)

|U(ρ, x, y, t)|.

Then, we replace the integrals by quadrature rules that we choose, as follows.∫e

he,K(x, y, t)vh(x, y)ds ≈L∑

l=1

ωlhe,K(xel, yel, t)vh(xel, yel)|e|,

∫K

f(ρ(x, y, t)) · ∇vh(x, y, t)dxdy ≈M∑

j=1

ωjf (ρ(xKj, yKj, t)) · ∇vh(xKj, yKj, t)|K|.

8

Finally, we obtain, for each element K ∈ Th, the weak formulation

d

dt

∫K

ρh(x, y, t)vh(x, y)dxdy +∑e∈∂K

L∑l=1

ωlhe,K(xel, yel, t)vh(xel, yel)|e|

−M∑

j=1

ωjf(ρ(xKj, yKj, t)) · ∇vh(xKj, yKj, t)|K| = 0, ∀vh ∈ V kh . (11)

This equation can be rewritten in ODE form as

d

dtρh = Lh(ρh), (12)

where the operator Lh is a discrete approximation of −∇ · f .

3.3 Time discretization

After discretizing in space by the DG method, we use the total variation diminishing (TVD)

Runge-Kutta methods developed in [13]; see also [32]. The second- and third-order versions

used in this paper are as follows. To solve the method of lines ODE,

ϕt = L(ϕ), (13)

the second-order TVD Runge-Kutta method is given by

ϕ(1) = ϕn + �tL(ϕn), (14)

ϕn+1 =1

2ϕn +

1

2ϕ(1) +

1

2�tL(ϕ(1)), (15)

and the third-order TVD Runge-Kutta method is given by

ϕ(1) = ϕn + �tL(ϕn), (16)

ϕ(2) =3

4ϕn +

1

4ϕ(1) +

1

4�tL(ϕ(1)), (17)

ϕn+1 =1

3ϕn +

2

3ϕ(2) +

2

3�tL(ϕ(2)). (18)

The TVD Runge-Kutta time discretization is just a convex combination of the Euler

forward steps. Then, we summarize the numerical procedure from density ρnh at time level tn

to ρn+1h at time level ρn+1

h into a complete algorithm in a Euler forward time discretization.

9

The numerical procedure to simulate the pedestrian model:

1. Solve the Eikonal equation (2) by the fast sweeping method described in Section 3.1, to

obtain φn and ∇φn.

2. Use the operator Lh(ρnh) in Section 3.2 to approximate −∇ · f in the conservation law

(1), where the flux is

fn =uρn

h∇φn

c= U(ρn

h)2ρnh∇φn. (19)

3. Then, we obtain ρn+1h = ρn

h + ΔtnLh(ρnh).

Time step Δtn = α hUmax

, where h is the minimum of the element diameters in Th, and

Umax is the maximum pedestrian walking speed. The constant α is dependent on the degree

of the polynomial in the DG method and the order of the Runge-Kutta method [19]. Then,

we obtain ρn+1h by use of the third-order TVD Runge-Kutta time discretization method.

4 The numerical tests

Consider a railway platform with an obstruction in the middle, as shown in Fig. 1. Pedes-

trians enter the platform from the left boundary Γi and leave at the right boundary Γo.

Initially, the platform is empty (ρ(x, 0) = 0, x ∈ Ω), and we set constants umax = 2 and

ρmax = 10 in equation (3). The flux f on boundary Γi is given by

f =

⎧⎪⎨⎪⎩(t/12, 0), 0 ≤ t ≤ 60,

(10 − t/12, 0), 60 ≤ t ≤ 120,

(0, 0), t ≥ 120.

The flux f on boundary Γo is given by equation (19), and f = 0 on boundary Γw. A similar

example has been studied in [7] with the WENO algorithm on a rectangular mesh. Our

motivation is to study this model on arbitrary geometries. In Fig. 2, we replace the square

obstruction with a circular obstruction that has the same barycenter and area.

Under the same initial conditions and boundary conditions, we compare the numerical

results in these two platforms. In both examples, we use the third-order DG method coupled

10

Γw

ΓoΓi

Obstruct

10m

20m

20m

40m 20m 40m

10m

10m

Figure 1: The railway platform with a square obstruction.

Γw

ΓoΓi

Γw

Obstruct

100m

50m

10m

10m

Figure 2: The railway platform with a circular obstruction.

with the second order fast sweeping method and the third order TVD Runge-Kutta method.

There are 3453 and 3417 elements in the triangulations of the square-obstacle and circle-

obstacle platforms, respectively, as showed in Fig. 3. For the square-obstacle platform, we

obtain the same results as those that are computed using the WENO scheme in [7].

In Fig. 4, we show density ρ in the two platforms at time t = 30, 60, 120, and 180. The

density in both platforms increases steadily and reaches the maximum near time t = 120.

In each platform, two shocks are seen clearly before the obstruction at time t = 120 due

11

X

Y

0 20 40 60 80 1000

10

20

30

40

50

(a) Triangulation of the square-obstacle platform.

X

Y

0 20 40 60 80 1000

10

20

30

40

50

(b) Triangulation of the circle-obstacle platform.

Figure 3: The triangulations of the square-obstacle and circle-obstacle platforms.

to a reduction in corridor width. Two triangular vacuum regions in front of and behind

the obstacles are also observed for the cost minimization strategy of pedestrians. At the

time near t = 240, the density in both platform returns to zero. As shown in Table 1,

the peak of the density in the circle-obstacle platform is less than the peak in the square-

obstacle platform at the same time, which means the circle-obstacle platform is superior to

the square-obstacle platform in discharging pedestrians more smoothly and efficiently.

12

Table 1: The maximum densities in the square-obstacle and circle-obstacle platforms at timet = 30, 60, 120, and 180.

t 30 60 120 180ρsquare−obstacle 1.80 5.53 8.99 6.92ρcircle−obstacle 1.44 4.65 8.55 6.32

Fig. 5 shows the cost potential φ at different times. The cost potential is dependent on

the density through the Eikonal equation (2), which depicts the instantaneous travel cost

from a given location to the exit of the platform. In conjunction with Fig. 4, we can see that

the greater the density, the closer the two contours, which is caused by the local travel cost

increasing as the density increases. We also give the flow flux f in Fig. 6. In conjunction

with Figs. 4 and 5, this shows the movement pattern of pedestrians, which is consistent with

equation (19).

5 Conclusion

The discontinuous Galerkin (DG) method, coupled with the fast sweeping method, has been

investigated to solve the model of pedestrian flows developed in [7] on arbitrary computa-

tional domains with triangular meshes. In this model, density ρ is governed by the scalar

two-dimensional conservation law, in which the flow flux f is implicitly dependent on density

ρ through the Eikonal equation. To solve the Eikonal equation efficiently, we use the fast

sweeping method. Although the DG method can be extended to an arbitrary high order, the

limitations of the first- and second-order fast sweeping method restrict the whole algorithm

to be no more than second order accuracy. In future work, we will explore a higher order

DG method with a higher order efficient method to solve the Eikonal equation on triangular

meshes.

13

6 Acknowledgments

The research of the second author was supported by a grant from the Research Grants Coun-

cil of the Hong Kong Special Administrative Region, China (HKU 7183/06E). The research

of third and fourth authors was supported by the National Natural Science Foundation of

China (grant 10671190). Additional support for the fourth author was provided by NSF

grant DMS-0510345. The research of the fifth author was supported by a grant from the

Research Grants Council of the Hong Kong Special Administrative Region, China (PolyU

5168/04E).

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

(a) ρ, t = 30.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

(b) ρ, t = 30.

0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8

(c) ρ, t = 60.

0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8

(d) ρ, t = 60.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

(e) ρ, t = 120.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

(f) ρ, t = 120.

0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 5.1 5.4 5.7 6 6.3

(g) ρ, t = 180.

0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 5.1 5.4 5.7 6 6.3

(h) ρ, t = 180.

Figure 4: Density ρ at different time t = 30, 60, 120, and 180 in the square-obstacle andcircle-obstacle platforms.

18

2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50

(a) φ, t = 30.

2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50

(b) φ, t = 30.

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60

(c) φ, t = 60.

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60

(d) φ, t = 60.

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78

(e) φ, t = 120.

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78

(f) φ, t = 120.

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51

(g) φ, t = 180.

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51

(h) φ, t = 180.

Figure 5: The cost potential φ at different time t = 30, 60, 120, and 180 in the square-obstacleand circle-obstacle platforms.

19

(a) f , t = 30. (b) f , t = 30.

(c) f , t = 60. (d) f , t = 60.

(e) f , t = 120. (f) f , t = 120.

(g) f , t = 180. (h) f , t = 180.

Figure 6: The flow flux f at different time t = 30, 60, 120, and 180 in the square-obstacleand circle-obstacle platforms.

20