WENO Scheme And Fast Sweeping Method And Applications In The Pedestrian Flows

In this thesis, we apply numerical methods, such as Weighted Essentially Non-Oscillatory (WENO) scheme, fast sweeping method, finite difference, Runge-Kutta time discretization, for reactive dynamic user equilibrium continuum model for the pedestrian flows in two-dimensional walking facility and study the follow-the-crowd effect in the reactive dynamic user equilibrium continuum model for the pedestrian flows. In the reactive dynamic user equilibrium model for the pedestrian flows based on the dynamic continuum model for the pedestrian flows proposed by Hughes in 2002, the pedestrian route choice strategy satisfies the reactive dynamic user equilibrium principle in which a pedestrian chooses a route to minimize the instantaneous travel cost to the destination. The pedestrian density, flux, and walking speed are governed by a two-dimensional scalar conservation law. The flux direction is implicitly dependent on the instantaneous global density through an Eikonal equation. In our numerical simulation, in each time step, we first use WENO Godunov fast sweeping method for the Eikonal equation, and obtain the flux, then use WENO finite difference scheme for the conservation law. The follow-the-crowd effect has been incorporated in the reactive dynamic user equilibrium model for the pedestrian flows, which constitutes a viscosity effect in the continuum model. Compared with the simple Eikonal equation, the flux direction is implicitly dependent on the pedestrian density through an more complicated Hamilton-Jacobi equation at the time. So, a pseudo time-dependent method is used for the Hamilton-Jacobi equation.High order fast sweeping methods have been developed recently in the literature to solve static Hamilton-Jacobi equations efficiently. Comparing with the first order fast sweeping methods, the high order fast sweeping methods arc more accurate, but they often require additional numerical boundary treatment for several grid points near the boundary because of the wider numerical stencil. It is particularly important to treat the points near the inflow boundary accurately, as the information would flow into the computational domain and would affect global accuracy. In the literature, the numerical solution at these boundary points are cither fixed with the exact solution, which is not always feasible, or computed with a first order discretization, which could reduce the global accuracy. In this thesis, we discuss two strategies to handle the inflow boundary conditions. One is based on the numerical solutions of a first order fast sweeping method with several different mesh sizes near the boundary and a Richardson extrapolation, the other is based on a Lax-Wendroff type procedure to repeatedly utilizing the PDE to write the normal spatial derivatives to the inflow boundary in terms of the tangential derivatives, thereby obtaining high order solution values at the grid points near the inflow boundary. We explore these two approaches using the fast sweeping high order WENO scheme in for solving the static Eikonal equation as a representative example. Numerical examples are given to demonstrate the performance of these two approaches.