Numerical Methods for Initial Value Problems in Ordinary Differential Equations

Preface


1 Preliminaries


1.1 The Difference Operators


1.2 Theory of Interpolation


1.3 Finite Difference Equations


1.4 Linear Systems with Constant Coefficients


1.5 Distribution of Roots of Polynomials


1.6 First Integral Mean Value Theorem


1.7 Common Norms in ODEs


2 Numerical Integration Algorithms


2.1 Introduction


2.2 Existence of Solution, Numerical Approach


2.3 Special IVPs


2.4 Error Propagation, Stability and Convergence of Discretization Methods


3 Theory of One-Step Methods


3.1 General Theory of One-Step Methods


3.2 The Euler Scheme, the Inverse Euler Schem and Richardson's Extrapolation


3.3 The Convergence of Euler's Scheme


3.4 The Trapezoidal Scheme


4 Runge-Kutta Processes


4.1 General Theory of Runge-Kutta Processes


4.2 The Explicit Two-Stage Process


4.3 Convergence and Stability of Two-Stage Explicit R-K Scheme


4.4 Matrix Representation of the R-K Processes


4.5 Error Estimation and Stepsize Selection in R-K Processes


4.6 Implicit and Semi-Implicit R-K Processes


4.7 Rosenbrock Methods


5 Linear Multistep Methods


5.1 Starting Procedure


5.2 Explicit Linear Multistep Methods


5.3 Implicit Linear Multistep Methods


5.4 Implementation of the Predictor-Corrector Formulas


5.5 General Theory of Linear Multistep Methods


5.6 Automatic Implementation of the Adams Scheme


6 Numerical Treatment of Singular/Discontinuous Initial Value Problems


6.1 Introduction


6.2 Non-Polynomial Methods


6.3 The Inverse Polynomial Methods


6.4 Local Error Estimates in Automatic Codes for Discontinuous Systems


7 Extrapolation Processes and Singularities


7.1 Introduction


7.2 Generation of the Zero-th Column of Extrapolation Table


7.3 Polynomial and Rational Extrapolation


7.4 Convergence and Stability Properties of Extrapolation Processes


7.5 Practical Implementation of Extrapolation Processes


8 Stiff Initial Value Problems


8.1 The Concept of Stiffness


8.2 Stiff and Nonstiff Algorithms


8.3 Solution of Nonlinear Equations and Estimation of Jacobians


8.4 Region of Absolute Stability


8.5 Stability Criteria for Stiff Methods


8.6 Stronger Stability Properties of IRK Processes


8.7 One-Leg Multistep Methods


9 Stiff Algorithms


9.1 What are Stiff Algorithms


9.2 Efficient Implementation of Implicit Runge-Kutta Methods


9.3 The Backward Differentiation Formula


9.4 Second Derivative Formulas


9.5 Extrapolation Processes for Stiff Systems


9.6 Mono-Implicit Runge-Kutta Methods


10 Second Order Differential Equations


10.1 Introduction


10.2 Linear Multistep Methods and the Concept of P-Stability


10.3 Derivation of P-Stable Formulas


10.4 One-Leg Multistep Methods for Second Order IVPs


10.5 Multiderivative Methods


11 Recent Developments in Ode Solvers

References