Numerical Methods for Ordinary Differential Equations

Abstract

Contents Preface to the first edition xiii Preface to the second edition xvii 1 Differential and Difference Equations 1 10 Differential Equation Problems 1 100 Introduction to differential equations 1 101 The Kepler problem 4 102 A problem arising from the method of lines 7 103 The simple pendulum 10 104 A chemical kinetics problem 14 105 The Van der Pol equation and limit cycles 16 106 The Lotka-Volterra problem and periodic orbits 18 107 The Euler equations of rigid body rotation 20 11 Differential Equation Theory 22 110 Existence and uniqueness of solutions 22 111 Linear systems of differential equations 24 112 Stiff differential equations 26 12 Further Evolutionary Problems 28 120 Many-body gravitational problems 28 121 Delay problems and discontinuous solutions 31 122 Problems evolving on a sphere 32 123 Further Hamiltonian problems 34 124 Further differential-algebraic problems 36 13 Difference Equation Problems 38 130 Introduction to difference equations 38 131 A linear problem 38 132 The Fibonacci difference equation 40 133 Three quadratic problems 40 134 Iterative solutions of a polynomial equation 41 135 The arithmetic-geometric mean 43 CONTENTS 14 Difference Equation Theory 44 140 Linear difference equations 44 141 Constant coefficients 45 142 Powers of matrices 46 Numerical Differential Equation Methods 51 20 The Euler Method 51 200 Introduction to the Euler methods 51 201 Some numerical experiments 54 202 Calculations with stepsize control 58 203 Calculations with mildly stiff problems 60 204 Calculations with the implicit Euler method 63 21 Analysis of the Euler Method 65 210 Formulation of the Euler method 65 211 Local truncation error 66 212 Global truncation error 66 213 Convergence of the Euler method 68 214 Order of convergence 69 215 Asymptotic error formula 72 216 Stability characteristics 74 217 Local truncation error estimation 79 218 Rounding error 80 22 Generalizations of the Euler Method 85 220 Introduction 85 221 More computations in a step 86 222 Greater dependence on previous values 87 223 Use of higher derivatives 88 224 Multistep-multistage-multiderivative methods 90 225 Implicit methods 91 226 Local error estimates 91 23 Runge-Kutta Methods 93 230 Historical introduction 93 231 Second order methods 93 232 The coefficient tableau 94 233 Third order methods 95 234 Introduction to order conditions 95 235 Fourth order methods 98 236 Higher orders 99 237 Implicit Runge-Kutta methods 99 238 Stability characteristics 100 239 Numerical examples 103 CONTENTS vii 24 Linear Multistep Methods 105 …