An efficient discontinuous Galerkin method on triangular meshes for a pedestrian flow model Yinhua Xia 1 , S.C. Wong 2 , Mengping Zhang 3 , Chi-Wang Shu 4 and William H.K. Lam 5 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 1 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China. E-mail: [email protected]. 2 Department of Civil Engineering, The University of Hong Kong, Hong Kong, China. E-mail: [email protected]. 3 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China. E-mail: [email protected]. 4 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail: [email protected]. 5 Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, China. E-mail: [email protected]. 1
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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: [email protected].
2Department of Civil Engineering, The University of Hong Kong, Hong Kong, China. E-mail:[email protected].
3Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China.E-mail: [email protected].
4Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail:[email protected].
5Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong,China. E-mail: [email protected].
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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-
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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
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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]).
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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-
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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