High-Order Computational Scheme for a Dynamic Continuum Model for Bi-Directional Pedestrian Flows

In this article, the authors 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. The dynamic continuum model for bi-directional pedestrian flows is reviewed first, which is composed of a coupled system of a conservation law and an Eikonal equation. Next, the authors 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. A comparison of the numerical results of the model from the first-order and high-order methods is provided, and it is concluded 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.

[1]  Burton Wendroff,et al.  A two-dimensional HLLE riemann solver and associated godunov-type difference scheme for gas dynamics☆ , 1999 .

[2]  Serge P. Hoogendoorn,et al.  Walking infrastructure design assessment by continuous space dynamic assignment modeling , 2004 .

[3]  Shing Chung Josh Wong,et al.  Combined distribution and assignment model for a continuum traffic equilibrium problem with multiple user classes , 2006 .

[4]  R. Newcomb VISCOSITY SOLUTIONS OF HAMILTON-JACOBI EQUATIONS , 2010 .

[5]  Shing Chung Josh Wong,et al.  A predictive dynamic traffic assignment model in congested capacity-constrained road networks , 2000 .

[6]  Danping Peng,et al.  Weighted ENO Schemes for Hamilton-Jacobi Equations , 1999, SIAM J. Sci. Comput..

[7]  Serge P. Hoogendoorn,et al.  DYNAMIC USER-OPTIMAL ASSIGNMENT IN CONTINUOUS TIME AND SPACE , 2004 .

[8]  Chi-Wang Shu,et al.  Fast Sweeping Fifth Order WENO Scheme for Static Hamilton-Jacobi Equations with Accurate Boundary Treatment , 2010, J. Sci. Comput..

[9]  Chi-Wang Shu HIGH ORDER NUMERICAL METHODS FOR TIME DEPENDENT HAMILTON-JACOBI EQUATIONS , 2007 .

[10]  Shing Chung Josh Wong AN ALTERNATIVE FORMULATION OF D'ESTE'S TRIP ASSIGNMENT MODEL , 1994 .

[11]  Mengping Zhang,et al.  Fifth order fast sweeping WENO scheme for static Hamilton-Jacobi equations with accurate boundary treatment , 2022 .

[12]  Hongkai Zhao,et al.  High Order Fast Sweeping Methods for Static Hamilton–Jacobi Equations , 2006, J. Sci. Comput..

[13]  Shinji Tanaka,et al.  Dynamic Cell Transmission–Based Pedestrian Model with Multidirectional Flows and Strategic Route Choices , 2007 .

[14]  Shing Chung Josh Wong,et al.  A review of the two-dimensional continuum modeling approach to transportation problems , 2006 .

[15]  Shing Chung Josh Wong,et al.  Finite element solution for the continuum traffic equilibrium problems , 1998 .

[16]  Shing Chung Josh Wong,et al.  An efficient discontinuous Galerkin method on triangular meshes for a pedestrian flow model , 2008 .

[17]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1996 .

[18]  Roger L. Hughes,et al.  A continuum theory for the flow of pedestrians , 2002 .

[19]  Shing Chung Josh Wong,et al.  HOUSING ALLOCATION PROBLEM IN A CONTINUUM TRANSPORTATION SYSTEM , 2007 .

[20]  Shing Chung Josh Wong,et al.  Multi-commodity traffic assignment by continuum approximation of network flow with variable demand , 1998 .

[21]  Chi-Wang Shu,et al.  Revisiting Hughes’ dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm , 2009 .

[22]  Shing Chung Josh Wong,et al.  Improved Solution Algorithm for Multicommodity Continuous Distribution and Assignment Model , 2004 .

[23]  Shing Chung Josh Wong,et al.  Cordon-Based Congestion Pricing in a Continuum Traffic Equilibrium System , 2005 .

[24]  Shing Chung Josh Wong,et al.  Dynamic continuum model for bi-directional pedestrian flows , 2009 .

[25]  Chi-Wang Shu,et al.  A REACTIVE DYNAMIC CONTINUUM USER EQUILIBRIUM MODEL FOR BI-DIRECTIONAL PEDESTRIAN FLOWS ∗ , 2009 .

[26]  Sze Chun Wong,et al.  A Continuous Equilibrium Model for Estimating Market Areas of Competitive Facilities with Elastic Demand and Market Externality , 2000, Transp. Sci..

[27]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[28]  Shing Chung Josh Wong,et al.  Determining Market Areas Captured by Competitive Facilities: A Continuous Equilibrium Modeling Approach , 1999 .

[29]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[30]  Hongkai Zhao,et al.  A fast sweeping method for Eikonal equations , 2004, Math. Comput..

[31]  Chi-Wang Shu Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[32]  Xu-Dong Liu,et al.  Positive schemes for solving multi-dimensional hyperbolic systems of conservation laws II , 2003 .

[33]  Serge P. Hoogendoorn,et al.  Pedestrian route-choice and activity scheduling theory and models , 2004 .

[34]  Shing Chung Josh Wong,et al.  Bi-directional pedestrian stream model , 2011 .