A Study of Some Numerical Schemes for Hyperbolic PDE Problems

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梁長雯(Chang-Wen Liang) 查詢紙本館藏

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數學系

論文名稱

(A Study of Some Numerical Schemes for Hyperbolic PDE Problems)

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摘要(中)

我們將討論一些基本的數值方法來解雙曲線偏微分方程(hyperbolic PDE equations)的問題，對於不同的數值方法分析其數值解的特性，探討各個數值方法的穩定性 (stability)、收斂性 (convergence) 及產生的數值消散作用 (artificial viscosity) 的大小。然而一個雙曲線問題的解若為不連續的，其數值結果的表現通常較為不精確。而the conservation element and solution element method （The CESE method）是一種較新的數值方法，我們比較此數值方法與基本數值方法的結果，說明the CESE method 的好處。

摘要(英)

We will discuss some basic numerical methods to solve hyperbolic partial differential equations
problem. We analyze the behavior of its numerical solutions for different numerical
methods and discuss the stability, convergence and size of the resulting artificial viscosity.
If the solution of hyperbolic partial differential equation problem is discontinuous,
the performance of numerical results is usually less accurate. However, the conservation
element and solution element method(The CESE method)is a newer numerical method,
we will compare the results of this numerical method with the basic numerical method to
illustrate the benefits of the CESE method.

關鍵字(中)

★ 雙曲線方程式

關鍵字(英)

論文目次

Tables viii
Figures ix
1 Introduction 1
2 Scalar wave model problem 2
2.1 The linear advection equation 2
2.2 The nonlinear scalar equation 3
3 Numerical schemes 5
3.1 The forward-in-time method 5
3.2 The backward-in-time method 7
3.3 Stability analysis 8
3.4 Domain of dependence 11
3.5 Artificial viscosity 13
4 Vector wave systems 17
4.1 The Lax-Friedrichs method 18
4.2 The Lax-Wendroff method 19
4.3 The CESE method 20
5 Numerical results 24
5.1 The linear equation case 24
5.2 The nonlinear equation case 37
5.3 The Euler equation 46
6 Conclusions 57
References 58

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指導教授

黃楓南

審核日期

2017-8-23

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