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