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Table of Contents
Intro
Preface
Contents
1 Ordinary Differential Equations
1.1 Initial Value Problems
1.2 Well-Posedness
1.3 Dissipative Problems
1.4 Conservative Problems
1.5 Stability of Solutions
1.6 Exercises
2 Discretization of the Problem
2.1 Domain Discretization
2.2 Difference Equations: The Discrete Counterpart of Differential Equations
2.2.1 Linear Difference Equations
2.2.2 Homogeneous Case
2.2.3 Inhomogeneous Case
2.3 Step-by-Step Schemes
2.4 A Theory of One-Step Methods
2.4.1 Consistency
2.4.2 Zero-Stability
2.4.3 Convergence
2.5 Handling Implicitness
2.6 Exercises
3 Linear Multistep Methods
3.1 The Principle of Multistep Numerical Integration
3.2 Handling Implicitness by Fixed Point Iterations
3.3 Consistency and Order Conditions
3.4 Zero-Stability
3.5 Convergence
3.6 Exercises
4 Runge-Kutta Methods
4.1 Genesis and Formulation of Runge-Kutta Methods
4.2 Butcher Theory of Order
4.2.1 Rooted Trees
4.2.2 Elementary Differentials
4.2.3 B-Series
4.2.4 Elementary Weights
4.2.5 Order Conditions
4.3 Explicit Methods
4.4 Fully Implicit Methods
4.4.1 Gauss Methods
4.4.2 Radau Methods
4.4.3 Lobatto Methods
4.5 Collocation Methods
4.6 Exercises
5 Multivalue Methods
5.1 Multivalue Numerical Dynamics
5.2 General Linear Methods Representation
5.3 Convergence Analysis
5.4 Two-Step Runge-Kutta Methods
5.5 Dense Output Multivalue Methods
5.6 Exercises
6 Linear Stability
6.1 Dahlquist Test Equation
6.2 Absolute Stability of Linear Multistep Methods
6.3 Absolute Stability of Runge-Kutta Methods
6.4 Absolute Stability of Multivalue Methods
6.5 Boundary Locus
6.6 Unbounded Stability Regions
6.6.1 A-Stability
6.6.2 Pad ̌Approximations
6.6.3 L-Stability
6.7 Order Stars
6.8 Exercises
7 Stiff Problems
7.1 Looking for a Definition
7.2 Prothero-Robinson Analysis
7.3 Order Reduction of Runge-Kutta Methods
7.4 Discretizations Free from Order Reduction
7.4.1 Two-Step Collocation Methods
7.4.2 Almost Collocation Methods
7.4.3 Multivalue Collocation Methods Free from Order Reduction
7.5 Stiffly-Stable Methods: Backward Differentiation Formulae
7.6 Principles of Adaptive Integration
7.6.1 Predictor-Corrector Schemes
7.6.2 Stepsize Control Strategies
7.6.3 Error Estimation for Runge-Kutta Methods
7.6.4 Newton Iterations for Fully Implicit Runge-Kutta Methods
7.7 Exercises
8 Geometric Numerical Integration
8.1 Historical Overview
8.2 Principles of Nonlinear Stability for Runge-Kutta Methods
8.3 Preservation of Linear and Quadratic Invariants
8.4 Symplectic Methods
8.5 Symmetric Methods
8.6 Backward Error Analysis
8.6.1 Modified Differential Equations
8.6.2 Truncated Modified Differential Equations
8.6.3 Long-Term Analysis of Symplectic Methods
8.7 Long-Term Analysis of Multivalue Methods
Preface
Contents
1 Ordinary Differential Equations
1.1 Initial Value Problems
1.2 Well-Posedness
1.3 Dissipative Problems
1.4 Conservative Problems
1.5 Stability of Solutions
1.6 Exercises
2 Discretization of the Problem
2.1 Domain Discretization
2.2 Difference Equations: The Discrete Counterpart of Differential Equations
2.2.1 Linear Difference Equations
2.2.2 Homogeneous Case
2.2.3 Inhomogeneous Case
2.3 Step-by-Step Schemes
2.4 A Theory of One-Step Methods
2.4.1 Consistency
2.4.2 Zero-Stability
2.4.3 Convergence
2.5 Handling Implicitness
2.6 Exercises
3 Linear Multistep Methods
3.1 The Principle of Multistep Numerical Integration
3.2 Handling Implicitness by Fixed Point Iterations
3.3 Consistency and Order Conditions
3.4 Zero-Stability
3.5 Convergence
3.6 Exercises
4 Runge-Kutta Methods
4.1 Genesis and Formulation of Runge-Kutta Methods
4.2 Butcher Theory of Order
4.2.1 Rooted Trees
4.2.2 Elementary Differentials
4.2.3 B-Series
4.2.4 Elementary Weights
4.2.5 Order Conditions
4.3 Explicit Methods
4.4 Fully Implicit Methods
4.4.1 Gauss Methods
4.4.2 Radau Methods
4.4.3 Lobatto Methods
4.5 Collocation Methods
4.6 Exercises
5 Multivalue Methods
5.1 Multivalue Numerical Dynamics
5.2 General Linear Methods Representation
5.3 Convergence Analysis
5.4 Two-Step Runge-Kutta Methods
5.5 Dense Output Multivalue Methods
5.6 Exercises
6 Linear Stability
6.1 Dahlquist Test Equation
6.2 Absolute Stability of Linear Multistep Methods
6.3 Absolute Stability of Runge-Kutta Methods
6.4 Absolute Stability of Multivalue Methods
6.5 Boundary Locus
6.6 Unbounded Stability Regions
6.6.1 A-Stability
6.6.2 Pad ̌Approximations
6.6.3 L-Stability
6.7 Order Stars
6.8 Exercises
7 Stiff Problems
7.1 Looking for a Definition
7.2 Prothero-Robinson Analysis
7.3 Order Reduction of Runge-Kutta Methods
7.4 Discretizations Free from Order Reduction
7.4.1 Two-Step Collocation Methods
7.4.2 Almost Collocation Methods
7.4.3 Multivalue Collocation Methods Free from Order Reduction
7.5 Stiffly-Stable Methods: Backward Differentiation Formulae
7.6 Principles of Adaptive Integration
7.6.1 Predictor-Corrector Schemes
7.6.2 Stepsize Control Strategies
7.6.3 Error Estimation for Runge-Kutta Methods
7.6.4 Newton Iterations for Fully Implicit Runge-Kutta Methods
7.7 Exercises
8 Geometric Numerical Integration
8.1 Historical Overview
8.2 Principles of Nonlinear Stability for Runge-Kutta Methods
8.3 Preservation of Linear and Quadratic Invariants
8.4 Symplectic Methods
8.5 Symmetric Methods
8.6 Backward Error Analysis
8.6.1 Modified Differential Equations
8.6.2 Truncated Modified Differential Equations
8.6.3 Long-Term Analysis of Symplectic Methods
8.7 Long-Term Analysis of Multivalue Methods