High-Order Computational Scheme for a Dynamic Continuum Model for Bi-Directional Pedestrian Flows
Tao Xiong
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
Search for more papers by this authorMengping Zhang
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
Search for more papers by this authorChi-Wang Shu
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
Search for more papers by this authorCorresponding Author
S.C. Wong
Department of Civil Engineering, The University of Hong Kong, Hong Kong, China
To whom correspondence should be addressed. E-mail: [email protected].Search for more papers by this authorPeng Zhang
Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China
Search for more papers by this authorTao Xiong
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
Search for more papers by this authorMengping Zhang
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
Search for more papers by this authorChi-Wang Shu
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
Search for more papers by this authorCorresponding Author
S.C. Wong
Department of Civil Engineering, The University of Hong Kong, Hong Kong, China
To whom correspondence should be addressed. E-mail: [email protected].Search for more papers by this authorPeng Zhang
Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China
Search for more papers by this authorAbstract
Abstract: In this article, we present a high-order weighted essentially non-oscillatory (WENO) scheme, coupled with a high-order fast sweeping method, for solving a dynamic continuum model for bi-directional pedestrian flows. We first review the dynamic continuum model for bi-directional pedestrian flows. This model is composed of a coupled system of a conservation law and an Eikonal equation. Then we present the first-order Lax–Friedrichs difference scheme with first-order Euler forward time discretization, the third-order WENO scheme with third-order total variation diminishing (TVD) Runge–Kutta time discretization, and the fast sweeping method, and demonstrate how to apply them to the model under study. We present a comparison of the numerical results of the model from the first-order and high-order methods, and conclude that the high-order method is more efficient than the first-order one, and they both converge to the same solution of the physical model.
REFERENCES
- Asano, M., Sumalee, A., Kuwanhara, M. & Tanaka, S. (2007), Dynamic cell-transmission-based pedestrian model with multidirectional flows and strategic route choices, Transportation Research Record, 2039, 42–9.
- Crandall, M. & Lions, P. L. (1983), Viscosity solutions of Hamilton-Jacobi equations, Transactions of American Mathematical Society, 277, 1–42.
-
Ho, H. W. &
Wong, S. C. (2006), Two-dimensional continuum modeling approach to transportation problems, Journal of Transportation Systems Engineering and Information Technology, 6(6), 53–72.
10.1016/S1570-6672(07)60002-6 Google Scholar
- Ho, H. W. & Wong, S. C. (2007), Housing allocation problem in a continuum transportation system, Transportmetrica, 3(1), 21–39.
- Ho, H. W., Wong, S. C. & Loo, B. P. Y. (2006), Combined distribution and assignment model for a continuum traffic equilibrium problem with multiple user classes, Transportation Research Part B, 40, 633–50.
- Ho, H. W., Wong, S. C., Yang, H. & Loo, B. P. Y. (2005), Cordon-based congestion pricing in a continuum traffic equilibrium system, Transportation Research Part A, 39, 813–34.
- Hoogendoorn, S. P. & Bovy, P. H. L. (2004a), Pedestrian route-choice and activity scheduling theory and models, Transportation Research Part B, 38, 169–90.
- Hoogendoorn, S. P. & Bovy, P. H. L. (2004b), Dynamic user-optimal assignment in continuous time and space, Transportation Research Part B, 38, 571–92.
- Hoogendoorn, S. P., Bovy, P. H. L. & Daamen, W. (2003), Walking infrastructure design assessment by continuous space dynamic assignment modeling, Journal of Advanced Transportation, 38, 69–92.
- Huang, L., Wong, S. C., Zhang, M., Shu, C.-W. & Lam, W. H. K. (2009a), Revisiting Hughes' dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm, Transportation Research Part B, 43, 127–41.
- Huang, L., Xia, Y., Wong, S. C., Shu, C.-W., Zhang, M. & Lam, W. H. K. (2009b), A dynamic continuum model for bi-directional pedestrian flows, Proceedings of the Institution of Civil Engineers, Engineering and Computational Mechanics, 162(2), 67–75.
- Hughes, R. L. (2002), A continuum theory for the flow of pedestrians, Transportation Research Part B, 36, 507–35.
- Jiang, G. & Peng, D. P. (2000), Weighted ENO schemes for Hamilton-Jacobi equations, SIAM Journal on Scientific Computing, 21, 2126–43.
- Jiang, G. & Shu, C.-W. (1996), Efficient implementation of weighted ENO schemes, Journal of Computational Physics, 126, 202–28.
- Jiang, Y. Q., Xiong, T., Wong, S. C., Shu, C.-W., Zhang, M., Zhang, P. & Lam, W. H. K. (2009), A reactive dynamic continuum user equilibrium model for bi-directional pedestrian flows, Acta Mathematica Scientia, 29(6), 1541–55.
- Liu, X. D. & Lax, P. D. (2003), Positive schemes for solving multi-dimensional hyperbolic systems of conservation laws II, Journal of Computational Physics, 187(2), 428–40.
- Liu, X. D., Osher, S. & Chan, T. (1994), Weighted essentially nonoscillatory schemes, Journal of Computational Physics, 115(1), 200–12.
- Shu, C.-W. (1998), Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in B. Cockburn, C. Johnson, C.-W. Shu, E. Tadmor, and A. Quarteroni (eds.), Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, volume 1697, Springer, Berlin , pp. 325–432.
-
Shu, C.-W. (2007), High-order numerical methods for time dependent Hamilton-Jacobi equations, in
S. S. Goh,
A. Ron &
Z. Shen (eds.), Mathematics and Computation in Imaging Science and Information Processing, Lecture Notes Series,
Institute for Mathematical Sciences, National University of Singapore
,
Singapore
, volume 11, World Scientific Press, pp. 47–91.
10.1142/9789812709066_0002 Google Scholar
- Shu, C.-W. & Osher, S. (1988), Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of Computational Physics, 77, 439–71.
- Tong, C. O. & Wong, S. C. (2000), A predictive dynamic traffic assignment model in congested capacity-constrained road networks, Transportation Research Part B, 34, 625–44.
- Wendroff, B. (1999), A two-dimensional HLLE Riemann solver and associated Godunov-type difference scheme for gas dynamics, Computers and Mathematics with Applications, 38, 175–85.
- Wong, S. C. (1994), An alternative formulation of D'Este's trip assignment model, Transportation Research Part B, 28, 187–96.
- Wong, S. C. (1998), Multi-commodity traffic assignment by continuum approximation of network flow with variable demand, Transportation Research Part B, 32, 567–81.
- Wong, S. C., Lee, C. K. & Tong, C. O. (1998), Finite element solution for the continuum traffic equilibrium problems, International Journal for Numerical Methods in Engineering, 43(7), 1253–73.
- Wong, S. C., Leung, W. L., Chan, S. H., Lam, W. H. K., Yung, N. H. C., Liu, C. Y. & Zhang, P. (2010), Bi-directional pedestrian stream model with oblique intersecting angle, ASCE Journal of Transportation Engineering, 136(3), 234–42.
- Wong, S. C. & Yang, H. (1999), Determining market areas captured by competitive facilities: a continuous equilibrium modeling approach, Journal of Regional Science, 39, 51–72.
- Wong, S. C., Zhou, C. W., Lo, H. K. & Yang, H. (2004), An improved solution algorithm for the multi-commodity continuous distribution and assignment model, ASCE Journal of Urban Planning and Development, 130, 14–23.
- Xia, Y., Wong, S. C., Zhang, M., Shu, C.-W. & Lam, W. H. K. (2008), An efficient discontinuous Galerkin method on triangular meshes for a pedestrian flow model, International Journal for Numerical Methods in Engineering, 76, 337–50.
- Xiong, T., Zhang, M., Zhang, Y.-T. & Shu, C.-W. (2010), Fast sweeping fifth order WENO scheme for static Hamilton-Jacobi equations with accurate boundary treatment, Journal of Scientific Computing, 45(1–3), 514–36.
- Yang, H. & Wong, S. C. (2000), A continuous equilibrium model for estimating market areas of competitive facilities with elastic demand and market externalities, Transportation Science, 34, 216–27.
- Zhang, Y. T., Zhao, H. K. & Qian, J. (2006), High order fast sweeping methods for static Hamilton-Jacobi equations, Journal of Scientific Computing, 29(1), 25–56.
- Zhao, H. K. (2005), A fast sweeping method for Eikonal equations, Mathematics of Computation, 74, 603–27.
