High Resolution Schemes For Solving Hamilton-Jacobi Equations And Nonconservative Hyperbolic Systems

This dissertation introduces high resolution schemes, such as finite difference and finite volume essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) schemes, and discontinuous Galerkin (DG) finite element method, for solving Hamilton-Jacobi equations and nonconservative hyperbolic systems, espe-cially nonconservative Euler equations in fluid dynamics, with applications to pedes-trian flow models and multi-phase flow problems. The dissertation is mainly divided into the following two parts:In the first part of the dissertation, we first extend the fast sweeping third order WENO scheme for solving static Hamilton-Jacobi equations to fast sweeping fifth or-der WENO scheme, including the Eikonal equations, which is a special class of static Hamilton-Jacobi equations. We apply the fast sweeping high order WENO scheme for solving the Eikonal equation coupled with the high order WENO scheme for solv-ing the hyperbolic conservation law equation, with Runge-Kutta time discretization, to numerically simulate the two-dimensional macroscopic reactive dynamic continuum pedestrian flow models. The macroscopic reactive dynamic continuum pedestrian flow models are based on the dynamic user equilibrium principle, which minimizes the in-stantaneous travel cost for a pedestrian choosing a route towards his or her destination, and contain a mass conservation law equation for the pedestrian flow density, cou-pled with an Eikonal equation of potential function for deciding the direction of the flow flux. Comparing the numerical results, we have found that high order numerical schemes with much coarser mesh sizes can efficiently obtain almost the same results for lower order scheme with much more refined mesh sizes. We also use the high order high resolution scheme to simulate the crossing pedestrian flow models, and we can obtain the explicit lane formation phenomenon, which is similar to the microscopic pedestrian flow models, but our macroscopic model is much more efficient and effec-tive. For time dependent Hamilton-Jacobi equations, ENO and WENO schemes, DG and LDG methods, all are very efficient high order high resolution schemes for solv- ing such type of equations. In this dissertation, with respect to the Hamilton-Jacobi equations, we give an optimal L2priori error estimate of the one-dimensional and two-dimensional DG and LDG methods for directly solving the time dependent Hamilton-Jacobi equations.In the second part of the dissertation, we use the high order finite volume WENO scheme with subcell resolution for computing the nonconservative Euler equations, with application to multi-phase flows. High resolution schemes, such as finite differ-ence and finite volume ENO and WENO schemes, can well solve the single component conservative Euler equations. However, for multicomponent conservative Euler equa-tions, these numerical schemes show strong oscillations around the material interfaces, which is inherent for most of the current classic numerical schemes. Therefore, if we consider the primitive variables for the nonconservative Euler equations, it provides a model better suited for computations of propagating material fronts for the multicom-ponent flow models, and can result in clean and monotonic solution profiles. A rigorous definition of weak solutions has been given to the nonconservative products with the choice of a family of paths. Many path conservative schemes based on the path conser-vative theory have been developed by C. Pares et al., but only applied to shallow water equations. The main problem for the path conservative scheme is how to define the path and how does the numerical solution converge to the correct solution. R. Abgrall and S. Kami in2010pointed out the limitations of such path conservative schemes for computing nonconservative Euler equations. These schemes can not catch the right shock positions. In this dissertation, we identify that, most non-oscillatory schemes, such as total variational diminishing (TVD) scheme, ENO and WENO schemes, they would smear around the discontinuities with transitional points not landing on the cor-rect shock profiles, and the definition of paths based on these transitional points at the discontinuities, could not give us the right shock integral path, and would lead to the wrong numerical solutions. Our basic idea is that we use the finite volume WENO scheme with subcell resolution, to sharpen the smeared solution profile and greatly re-duce the transitional points, and we also extend the polynomials from the two adjacent cells to compute the limiting values at the discontinuous cell boundaries. With these boundary values, we can form a Riemann problem for conservative Euler equations. We use the exact Riemann solutions as the integral paths in the discontinuous cells, which would be a much more accurate path conservative scheme. Numerical exper-iments in one-dimensional Euler equations for one and two medium flows show the efficiency of our new approach.