| Preface Outline 1 Time-Dependent Processes in Science and Engineering 1.1 Newton's Celestial Mechanics 1.2 Classical Molecular Dynamics 1.3 Chemical Reaction Kinetics 1.4 Electrical Circuits Exercises 2 Existence and Uniqueness for Initial Value Problems 2.1 Global Existence and Uniqueness 2.2 Examples of Maximal Continuation 2.3 Structure of Nonunique Solutions 2.4 Weakly Singular Initial Value Problems 2.5 Singular Perturbation Problems 2.6 Quasilinear Differential-Algebraic Problems Exercises 3 Condition of Initial Value Problems 3.1 Sensitivity Under Perturbations 3.1.1 Propagation Matrices 3.1.2 Condition Numbers 3.1.3 Perturbation Index of DAE Problems 3.2 Stability of ODEs 3.2.1 Stability Concept 3.2.2 Linear Autonomous ODEs 3.2.3 Stability of Fixed Points 3.3 Stability of Recursive Mappings 3.3.1 Linear Autonomous Recursions 3.3.2 Spectra of Rational Matrix Functions Exercises 4 One-Step Methods for Nonstiff IVPs 4.1 Convergence Theory 4.1.1 Consistency 4.1.2 Convergence 4.1.3 Concept of Stiffness 4.2 Explicit Runge-Kutta Methods 4.2.1 Concept of Runge-Kutta Methods 4.2.2 Classical Runge-Kutta Methods 4.2.3 Higher-Order Runge-Kutta Methods 4.2.4 Discrete Condition Numbers 4.3 Explicit Extrapolation Methods 4.3.1 Concept of Extrapolation Methods 4.3.2 Asymptotic Expansion of Discretization Error 4.3.3 Extrapolation of Explicit Midpoint Rule 4.3.4 Extrapolation of StSrmer/Verlet Discretization Exercises 5 Adaptive Control of One-Step Methods 5.1 Local Accuracy Control 5.2 Control-Theoretic Analysis 5.2.1 Excursion to PID Controllers 5.2.2 Step-size Selection as Controller 5.3 Error Estimation 5.4 Embedded Runge-Kutta Methods 5.5 Local Versus Achieved Accuracy Exercises 6 One-Step Methods for Stiff ODE and DAE IVPs 6.1 Inheritance of Asymptotic Stability 6.1.1 Rational Approximation of Matrix Exponential 6.1.2 Stability Domains 6.1.3 Stability Concepts 6.1.4 Reversibility and Discrete Isometries …… 7 Multistep Methods for ODE and DAE IVPs 8 Boundary Value Problems for ODEs References Software Index |
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