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Discontinuous Galerkin Methods for Hyperbolic Systems of PDEs and the Bathtub Model for Traffic Flow


Macroscopic models for flows strive to depict the physical world by considering quantities of interest at the aggregate level versus focusing on discrete particles. This thesis contains three major sections which construct numerical methods for macroscopic flow models.

In the first part, we develop local discontinuous Galerkin (LDG) methods for the Boussinesq coupled BBM system. The system contains nonlinear dispersive wave equations, containing nonlinearities and dispersive effects. We provide two choices of numerical fluxes, namely the upwind and alternating fluxes, and establish their stability analysis. The error estimate for the linearized BBM system is carried out with the alternating flux. A time discretization conserving the Hamiltonian numerically using the midpoint rule with a nontrivial nonlinear term in the discretization is proposed. Numerical examples demonstrate the accuracy, long-time simulation, and Hamiltonian conservation properties of the proposed method.

In the second part, we consider the second-order Aw-Rascle (AR) traffic flow model on a network, and propose a discontinuous Galerkin (DG) method for solving the AR system of hyperbolic PDEs with appropriate coupling conditions at the junctions. On each road, the standard DG method with Lax-Friedrichs flux is employed, and at the junction, we solve an optimization problem to evaluate the numerical flux of the DG method. As the choice of well-posed coupling conditions for the AR model is not unique, we test different coupling conditions at the junctions. Numerical examples are provided to demonstrate the high-order accuracy, and comparison of results between the first-order Lighthill-Whitham-Richards model and the second-order AR model. The ability of the model to capture the capacity drop phenomenon is also explored.

In the third part, we consider the basic bathtub model of traffic congestion. The model has an unfamiliar mathematical structure, with a delay differential equation with an endogenous delay at its core. Early papers on the model circumvented this complication by making approximating assumptions, but without solution of the proper model it is unclear how accurate the results are. We develop a customized method for computational solution of equilibrium in the basic bathtub model with smooth preferences that exploits the mathematical structure of the problem.

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