微分方程數(shù)值方法引論

內(nèi)容概要

　　本書內(nèi)容包括：初值問題、兩點邊界值問題、擴(kuò)散問題、平流方程、橢圓型問題等。

書籍目錄

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

編輯推薦

　　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.

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