摘要
本文主要研究高分辨率方法,如高阶有限差分和有限体积加权本质无振荡格式(即：WENO格式)以及间断Gerlerkin有限元方法(即：DG方法),用于求解哈密顿-雅可比方程和非守恒双曲方程组,特别是流体力学中的非守恒欧拉方程组；并成功地应用于行人流问题和多相流问题。论文主要分为如下两大部分：
论文的第一大部分,我们推广了求解静态哈密顿-雅可比方程中的快速扫描三阶WENO格式至快速扫描五阶WENO格式,其中包含静态哈密顿-雅可比方程中一类特殊的方程,Eikonal方程。结合快速扫描高阶WENO格式求解Eikonal方程,高阶WENO格式求解双曲守恒律方程,以及Runge-Kutta时间离散,我们对二维宏观动态反应连续行人流模型进行了高阶数值模拟。宏观动态反应连续行人流模型是基于动态反应行人平衡原则,在此原则下行人选择行进路线,使得到达目的地的瞬时行进成本最小。它的控制方程为行人流密度的质量守恒双曲方程以及决定流通量方向的势函数的Eikonal方程的耦合方程组。通过数值模拟比较,高阶格式在粗网格下能够更有效的得出相应低阶格式细网格下的结果。而且我们用高阶高分辨率方法模拟宏观相向行人流模型,得出了明显的相向行人流分层现象,相比于微观模型下的行人流分层现象,宏观模型能更加有效更加快速的得到这样的结果。对于依赖时间的哈密顿-雅可比方程,本质无振荡格式(即：ENO格式)和WENO格式,以及DG方法和局部间断Galerkin有限元方法(即：LDG方法),都是求解此类方程的非常有效的高阶高分辨率方法。本文中,围绕哈密顿-雅可比方程,我们给出了直接求解一维和二维依赖时间的哈密顿-雅可比方程的DG方法和LDG方法的最优L2先验误差估计。
论文的第二大部分,我们用高阶有限体积WENO格式和次区间分辨技巧求解非守恒欧拉方程组,并应用于多相流问题。高分辨率方法如有限差分和有限体积高阶ENO格式和WENO格式,都能够很好的求解单介质的守恒欧拉方程组。但对于多介质的守恒欧拉方程组,高分辨率方法在介质界面处有很强的振荡,这种振荡是目前几乎所有经典格式都本质存在的。因此对于这种多介质问题,如果考虑原始变量的非守恒方程组,它能够更好的模拟多介质界面移动的问题,并能够得到精准单调的解。对于包含非守恒乘积项的方程组,严格的弱解是定义在一组积分路径上的。基于路径积分理论,C. Pares等人发展了一系列的路径守恒格式,但这些格式大部分只应用于求解浅水方程。路径守恒格式主要的问题是如何得到正确的积分路径,并确保数值解能够收敛到正确的解。R.Abgrall和S.Karni在2010年指出这种路径守恒格式在求解非守恒欧拉方程组时存在着问题,路径守恒格式收敛不到正确的激波解。本文中,我们指出大部分本质无振荡格式,如全变差递减格式(即：TVD格式),ENO和WENO格式,它们在间断处都会被抹平。这种抹平,使得格式的解在间断处存在过渡点,并且这种过渡点没有落在真正的间断积分路径上。因此路径守恒格式通过这些过渡点来构造积分曲线,不能够得到正确的积分路径,从而得不到正确的数值解。我们的做法是采用有限体积WENO格式和次区间分辨技巧,尽可能减少间断面的过渡点,来磨尖数值格式在间断面处的解。我们通过相邻区间的近似多项式延伸来得到数值格式在包含间断的区间里间断面处两端的点值,然后求解这组端点值满足的守恒欧拉方程组的黎曼问题。最后我们用该黎曼问题的真解作为包含间断的区间间断面处的积分路径,从而可以得到比较正确的路径守恒格式。一维欧拉方程组的单介质和多介质流问题的数值试验验证了我们新方法的有效性。
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.
引文
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