Designed for those people who want to gain a practical knowledge of modern techniques, this book contains all the material necessary for a course on the numerical solution of differential equations. Written by two of the field’s leading authorities, it provides a unified presentation of initial Value and boundary value problems in ODEs as well as differential-algebraic equations. The approach is aimed at a thorough understanding of the issues and methods for practical computation while avoiding an extensive theorem-proof type of exposition. It also addresses reasons why existing software succeeds or fails. This book is a practical and mathematically well informed introduction that emphasizes basic methods and theory, issues in the use and development of mathematical software, and examples from scientific engineering applications. Topics requiring an extensive amount of mathematical development, such as symplectic methods for Hamiltonian systems, are introduced, motivated, and included in the exercises, but a complete and rigorous mathematical presentation is referenced rather than included. This book is appropriate for senior undergraduate or beginning graduate students with a computational focus and practicing engineers and scientists who want to learn about computational differential equations. A beginning course in numerical analysis is needed, and a beginning course in ordinary differential equations Would be helpful.
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目录
List of Figures List of Tables Preface Part I: Introduction 1 Ordinary Differential Equations 1.1 IVPs 1.2 BVPs 1.3 Differential-Algebraic Equations 1.4 Families of Application Problems 1.5 Dynamical Systems 1.6 Notation Part II: Initial Value Problems 2 On Problem Stability 2.1 Test Equation and General Definitions 2.2 Linear，Constant-Coefficient Systems 2.3 Linear，Variable-Coefficient Systems 2.4 Nonlinear Problems 2.5 Hamiltonian Systems 2.6 Notes and References 2.7 Exercises 3 Basic Methods，Basic Concepts 3.1 A Simple Method: Forward Euler 3.2 Convergence，Accuracy，Consistency，and O-Stability 3.3 Absolute Stability 3.4 Stiffness: Backward Euler 3.4.1 Backward Euler 3.4.2 Solving Nonlinear Equations 3.5 A-Stability，Stiff Decay 3.6 Symmetry: Trapezoidal Method 3.7 Rough Problems 3.8 Software，Notes，and References 3.8.1 Notes 3.8.2 Software 3.9 Exercises 4 One-Step Methods 4.1 The First Runge-Kutta Methods 4.2 General Formulation of Runge-Kutta Methods 4.3 Convergence，O-Stability，and Order for Runge-Kutta Methods 4.4 Regions of Absolute Stability for Explicit Runge-Kutta Methods 4.5 Error Estimation and Control 4.6 Sensitivity to Data Perturbations 4.7 Implicit Runge-Kutta and Collocation Methods 4.7.1 Implicit Runge-Kutta Methods Based on Collocation 4.7.2 Implementation and Diagonally Implicit Methods 4.7.3 Order Reduction 4.7.4 More on Implementation and Singly Implicit Runge-Kutta Methods 4.8 Software，Notes，and References 4.8.1 Notes 4.8.2 Software 4.9 Exercises 5 Linear Multistep Methods 5.1 The Most Popular Methods 5.1.1 Adams Methods 5.1.2 BDF 5.1.3 Initial Values for Multistep Methods 5.2 Order，O-Stability，and Convergence 5.2.1 Order 5.2.2 Stability: Difference Equations and the Root Condition 5.2.3 O-Stability and Convergence 5.3 Absolute Stability 5.4 Implementation of Implicit Linear Multistep Methods 5.4.1 Functional Iteration 5.4.2 Predictor-Corrector Methods 5.4.3 Modified Newton Iteration 5.5 Designing Multistep Ceneral-Purpose Software 5.5.1 Variable Step-Size Formulae 5.5.2 Estimating and Controlling the Local Error 5.5.3 Approximating the Solution at Off-Step Points 5.6 Software，Notes，and References 5.6.1 Notes 5.6.2 Software 5.7 Exercises Part III: Boundary Value Problems 6 More Boundary Value Problem Theory and Applications 6.1 Linear BVPs and Green's Function 6.2 Stability of BVPs 6.3 BVP Stiffness 6.4 Some Reformulation Tricks 6.5 Notes and References 6.6 Exercises 7 Shooting 7.1 Shooting: A Simple Method and Its Limitations 7.1.1 Difficulties 7.2 Multiple Shooting 7.3 Software，Notes，and References 7.3.1 Notes 7.3.2 Software 7.4 Exercises 8 Finite Difference Methods for Boundary Value Problems 8.1 Midpoint and Trapezoidal Methods 8.1.1 Solving Nonlinear Problems: Quasi-Linearization 8.1.2 Consistency，O-Stability，and Convergence 8.2 Solving the Linear Equations 8.3 Higher-Order Methods 8.3.1 Collocation 8.3.2 Acceleration Techniques 8.4 More on Solving Nonlinear Problems 8.4.1 Damped Newton 8.4.2 Shooting for Initial Guesses 8.4.3 Continuation 8.5 Error Estimation and Mesh Selection 8.6 Very Stiff Problems 8.7 Decoupling 8.8 Software，Notes，and References 8.8.1 Notes 8.8.2 Software 8.9 Exercises Part IV: Differential-Algebraic Equations 9 More on Differential-Algebraic Equations 9.1 Index and Mathematical Structure 9.1.1 Special DAE Forms 9.1.2 DAE Stability 9.2 Index Reduction and Stabilization: ODE with Invariant 9.2.1 Reformulation of Higher-Index DAEs 9.2.2 ODEs with Invariants 9.2.3 State Space Formulation 9.3 Modeling with DAEs 9.4 Notes and References 9.5 Exercises 10 Numerical Methods for Differential-Algebraic Equations 10.1 Direct Discretization Methods 10.1.1 A Simple Method: Backward Euler 10.1.2 BDF and General Multistep Methods 10.1.3 Radau Collocation and Implicit Runge-Kutta Methods 10.1.4 Practical Difficulties 10.1.5 Specialized Runge-Kutta Methods for Hessenberg Index-2 DAEs 10.2 Methods for ODEs on Manifolds 10.2.1 Stabilization of the Discrete Dynamical System 10.2.2 Choosing the Stabilization Matrix F 10.3 Software，Notes，and References 10.3.1 Notes 10.3.2 Software 10.4 Exercises Bibliography Index
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