CPR Method With Limiters And Study On Traffic Flow Models

This dissertation is concerned with numerical methods for partial differential equa-tions (PDEs), such as conservation law (CL) and Hamilton-Jacobi (HJ) equation, and their applications in traffic flow models. We can mainly divide the dissertation into the following two parts:In the first part, we study the correction procedure via reconstruction (CPR) frame-work for solving hyperbolic conservation laws. Since the CPR method is a high-order linear scheme, it may generate spurious oscillations for problems containing strong dis-continuities. We adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for discontinuous Galerkin (DG) schemes, to the CPR framework on both structured and unstructured meshes with straight or curved edges. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the CPR framework in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt in this paper is particularly simple to implement and will not harm the conservativeness of the CPR framework. Also, it uses information only from the target cell and its immediate neighbors, thus can main-tain the compactness of the CPR framework. An important property of the entropy solution of a scalar conservation law is that it satisfies a strict maximum principle. In particular, the solution will preserve positive if the initial value is positive. In practice, some physical quantities should be positive, such as the density and pressure in Euler equations for compressible gas dynamics. Since the CPR framework with the WENO limiter does not in general preserve positivity of the solution in above situations, we also extend the positivity-preserving limiters originally designed for DG scheme to the CPR framework. Numerical results in one and two dimensions are provided to illustrate the good behavior of these limiters.In the second part, we focus on the modeling and numerical methods for dynamic traffic flow problem in both isotropic and anisotropic cases. For the isotropic case, we revisit the predictive continuum dynamic user-equilibrium model proposed by Jiang et al.[54], since their model is not well posed due to an inconsistency in the route-choice strategy. We construct a new path-choice strategy and an improved model for a dense urban city that is arbitrary in shape and has a single central business district (CBD). For the anisotropic case, Hoogendoorn and Bovy [45] developed an approach for a pedes- trian user-optimal dynamic assignment. Although their model is very general, only an isotropic application example was given in their paper. We claim that the Hamilton-Jacobi-Bellman (HJB) equation in their model is difficult to compute numerically in an anisotropic case. To overcome this, we reformulate their model for a dense urban city with multiple CBDs. In our model, the HJB equation reduces to an HJ equation for easier computation. Both the isotropic and anisotropic models constructed in this dissertation consist of a CL to govern the density, in which the flow direction is de-termined by the path-choice strategy, and an HJ equation to compute the total travel cost. We apply robust numerical schemes to solve CL and HJ equations. Unlike other problems in which the initial conditions of both equations are set at the beginning of the modeling period, these two equations in our models have different computational directions in time and therefore cannot be solved together as usual. The simultaneous satisfaction of both equations can be treated as a fixed-point problem. A self-adaptive method of successive averages (MSA) is proposed to solve this fixed-point problem. This method can automatically determine the optimal MSA step size using the least squares approach. In the model for anisotropic case, we also need to solve a minimum value problem. We propose a simple way to solve it numerically. Numerical exam-ples for both isotropic and anisotropic cases are shown to demonstrate the effectiveness of the models and the solution algorithms. We also give comparisons of the models, solution algorithms and numerical results between the isotropic and anisotropic cases.