Numerical Methods for Ordinary Differential Equations

Abstract

Preface to the first edition xiii Preface to the second edition xvii 1 Differential and Difference Equations 1 10 Differential Equation Problems 100 Introduction to differential equations 101 The Kepler problem 102 A problem arising from the method of lines 103 The simple pendulum 104 A chemical kinetics problem 105 The Van der Pol equation and limit cycles 106 The Lotka-Volterra problem and periodic orbits 107 The Euler equations of rigid body rotation 11 Differential Equation Theory 110 Existence and uniqueness of Solutions 111 Linear Systems of differential equations 112 Stiff differential equations 12 Further Evolutionary Problems 120 Many-body gravitational problems 121 Delay problems and discontinuous Solutions 122 Problems evolving on a sphere 123 Further Hamiltonian problems 124 Further differential-algebraic problems 13 Difference Equation Problems 130 Introduction to difference equations 131 A linear problem 132 The Fibonacci difference equation 133 Three quadratic problems 134 Iterative Solutions of a polynomial equation 135 The arithmetic-geometric mean vi CONTENTS 14 Difference Equation Theory 140 Linear difference equations 141 Constant coefficients 142 Powers of matrices 2 Numerical Differential Equation Methods 20 The Euler Method 200 Introduction to the Euler methods 201 Some numerical experiments 202 Calculations with stepsize control 203 Calculations with mildly stiff problems 204 Calculations with the implicit Euler method 21 Analysis of the Euler Method 210 Formulation of the Euler method 211 Locol truncation error 212 Global truncation error 213 Convergence of the Euler method 214 Order of convergence 215 Asymptotic error formula 216 Stability characteristics 217 Locol truncation error estimation 218 Rounding error 22 Generalizations of the Euler Method 220 Order conditions for scalar problems 317 Independence of elementary weights 318 Locol truncation error 319 Global truncation error 32 Low Order Explicit Methods 320 Methods of orders less than 4 321 Simplifying assumptions 322 Methods of order 4 323 New methods from old 324 Order barriers 325 Methods of order 5 326 Methods of order 6 321 Methods of orders greater than 6 33 Runge-Kutta Methods with Error Estimates 330 Introduction 331 Richardson error estimates 332 Methods with built-in estimates 333 A class of error-estimating methods 334The methods of Fehlberg 335The methods of Verner 336The methods of Dormand and Prince 34 Implicit Runge-Kutta Methods

Efficient time-symmetric simulation of torqued rigid bodies using Jacobi elliptic functions

“…Some extra computational work is required to impose initial conditions. We also refer to [3] and [14] for related literature. In this paper we show how this approach is very competitive and we discuss the details of its further use in problems of rigid bodies subject to external forces.…”

Richardson Extrapolation

“…One-step methods are discussed in detail, for example, in Burrage (1995), Butcher (2003), Hairer, Nørsett and Wanner (1987), Henrici (1968), and Lambert (1991). Roughly speaking, only the approximation − of the solution at the grid-point − is used in the calculation of the approximation at the next grid-point of (1.6) when one-step methods are selected.…”

“…Therefore a reference solution was firstly obtained by solving the problem with a very small time-stepsize and a numerical method of high order. Actually, a threestage fifth-order fully-implicit Runge-Kutta algorithm, see Butcher (2003) or Hairer and Wanner (1991), was used with = and ≈ . − to calculate the reference solution.…”

A review of numerical methods for nonlinear partial differential equations

“…The solution of such semidiscrete systems is obtained by standard ODE solvers [82,96,97,28]. The finite-difference schemes (3.2)-(3.8) are typical examples of finite-difference approximations of nonlinear PDEs.…”