微分方程数值方法引论

Preface

1 Initial Value Problems

1.1 Introduction

1.1.1 Examples of IVPs

1.2 Methods Obtained from Numerical Differentiation .

1.2.1 The Five Steps

1.2.2 Additional Difference Methods

1.3 Methods Obtained from Numerical Quadrature

1.4 Runge--Kutta Methods

1.5 Extensions and Ghost Points

1.6 Conservative Methods

1.6.1 Velocity Verlet

1.6.2 Symplectic Methods

1.7 Next Steps

Exercises

2 Two-Point Boundary Value Problems

2.1 Introduction

2.1.1 Birds on a Wire

2.1.2 Chemical Kinetics

2.2 Derivative Approximation Methods

2.2.1 Matrix Problem

2.2.2 Tridiagonal Matrices

2.2.3 Matrix Problem Revisited

2.2.4 Error Analysis

2.2.5 Extensions

2.3 Residual Methods

2.3.1 Basis Functions

2.3.2 Residual

2.4 Shooting Methods

2.5 Next Steps

Exercises

3 Diffusion Problems

3.1 Introduction

3.1.1 Heat Equation

3.2 Derivative Approximation Methods

3.2.1 Implicit Method

3.2.2 Theta Method

3.3 Methods Obtained from Numerical Quadrature

3.3.1 Crank-Nicolson Method

3.3.2 L-Stability

3.4 Methods of Lines

3.5 Collocation

3.6 Next Steps

Exercises

4 Advection Equation

4.1 Introduction

4.1.1 Method of Characteristics

4.1.2 Solution Properties

4.1.3 Boundary Conditions

4.2 First-Order Methods

4.2.1 Upwind Scheme

4.2.2 Downwind Scheme

4.2.3 blumericul Domu'm of Dependence

4.2.4 Stability

4.3 Improvements

4.3.1 Lax-Wendroff Method

4.3.2 Monotone Methods

4.3.3 Upwind Revisited

4.4 Implicit Methods

Exercises

5 Numerical Wave Propagation

5.1 Introduction

5.1.1 Solution Methods

5.1.2 Plane Wave Solutions

5.2 Explicit Method

5.2.1 Diagnostics

5.2.2 Numerical Experiments

5.3 Numerical Plane Waves

5.3.1 Numerical Group Velocity

5.4 Next Steps

Exercises

6 Elliptic Problems

6.1 Introduction

6.1.1 Solutions

6.1.2 Properties of the Solution

6.2 Finite Difference Approximation

6.2.1 Building the Matrix

6.2.2 Positive Definite Matrices

6.3 Descent Methods

6.3.1 Steepest Descent Method

6.3.2 Conjugate Gradient Method

6.4 Numerical Solution of Laplace's Equation

6.5 Preconditioned Conjugate Gradient Method

6.6 Next Steps

Exercises

A Appendix

A.1 Order Symbols

A.2 Taylor's Theorem

A.3 Round-Off Error

A.3.1 Fhnction Evaluation

A.3.2 Numerical Differentiation

A.4 Floating-Point Numbers

References

Index

《微分方程数值方法引论》内容包括：初值问题、两点边界值问题、扩散问题、平流方程、椭圆型问题等。

This book shows how to derive.tcst and analyze numerical methods for solving differential equations，including both ordinarv and partial differential equations.The objective is that students learn to solve differential equations numerically and understand the mathematical and computational issues that arise when this iS done.Includes an extensive collection of exercises.which develop both the analytical and computational aspects ofthe material.In addition to more than 1 00 illustrations，the book includes a large collection of supplemental material：exercise sets.MATLAB computer codes for bOth student and instructor.1ecture slides and movies.