Chequerboard Modes and the Generalized Lax-Friedrichs Schemes for Hyperbolic Conservation Laws

论文摘要

用广义Lax-Friedrichs格式计算双曲守恒律时可能会出现棋盘波，本文对存在的棋盘波进行了分析。这个分析有助于认识和理解高频波对数值解的影响，从而较好地解释了当初始数据离散得不恰当时数值解中出现的振荡。这篇论文的主要贡献在于用有限傅利叶级数的知识阐明了在初始数据离散化的过程中出现了棋盘波，并用修正方程对棋盘波的变化进行了分析，找到了产生振荡的原因。

论文目录

• 摘要
• Abstract
• 1 Introduction
• 1.1 Backgrounds
• 1.2 Structure of the thesis
• 1.3 Preliminaries and notations
• 1.3.1 The chequerboard mode
• 1.3.2 Modified equation analysis
• 2 Local oscillations in the solutions by the generalized Lax-Friedrichs schemes
• 2.1 Examples of the linear hyperbolic conservation laws
• 2.2 Examples of the nonlinear hyperbolic conservation laws
• 3 Chequerboard modes in the initial discrerization
• 3.1 Single square signal case
• 3.2 Discretization of general initial data
• 4 Analysis of the evolution of the chequerboard mode by the generalized Lax-Friedrichs schemes
• 4.1 Modified equation analysis for the linear equations
• 4.2 Conlusions
• 5 Two spatial dimensions
• 5.1 Numerical experiments
• 5.2 Discrete Fourier analysi
• 5.2.1 Single square signal case
• 5.2.2 For the Linear advection equation
• 5.3 Modified equation analysis
• Bibliography
• ACKNOWLEDGEMENT