Numerical methods for ordinary differential equations

In recent years the study of numerical methods for solving ordinary differential equations has seen many new developments. This second edition of the author's pioneering text is fully revised and updated to acknowledge many of these developments. It includes a complete treatment of linear multistep methods whilst maintaining its unique and comprehensive emphasis on Runge-Kutta methods and general linear methods. Although the specialist topics are taken to an advanced level, the entry point to the volume as a whole is not especially demanding. Early chapters provide a wide-ranging introduction to differential equations and difference equations together with a survey of numerical differential equation methods, based on the fundamental Euler method with more sophisticated methods presented as generalizations of Euler. Features of the book include Introductory work on differential and difference equations. A comprehensive introduction to the theory and practice of solving ordinary differential equations numerically. A detailed analysis of Runge-Kutta methods and of linear multistep methods. A complete study of general linear methods from both theoretical and practical points of view. The latest results on practical general linear methods and their implementation. A balance between informal discussion and rigorous mathematical style. Examples and exercises integrated into each chapter enhancing the suitability of the book as a course text or a self-study treatise. Written in a lucid style by one of the worlds leading authorities on numerical methods for ordinary differential equations and drawing upon his vast experience, this new edition provides an accessible and self-contained introduction, ideal for researchers and students following courses on numerical methods, engineering and other sciences. Cover......Page 1 Numerical Methods for Ordinary Differential Equations (Second Edition)......Page 4 Copyright......Page 5 Contents......Page 6 Preface to the first edition......Page 14 Preface to the second edition......Page 18 100 Introduction to differential equations......Page 22 101 The Kepler problem......Page 25 102 A problem arising from the method of lines......Page 28 103 The simple pendulum......Page 31 104 A chemical kinetics problem......Page 35 105 The Van der Pol equation and limit cycles......Page 37 106 The Lotka–Volterra problem and periodic orbits......Page 39 107 The Euler equations of rigid body rotation......Page 41 110 Existence and uniqueness of solutions......Page 43 111 Linear systems of differential equations......Page 45 112 Stiff differential equations......Page 47 120 Many-body gravitational problems......Page 49 121 Delay problems and discontinuous solutions......Page 52 122 Problems evolving on a sphere......Page 53 123 Further Hamiltonian problems......Page 55 124 Further differential-algebraic problems......Page 57 131 A linear problem......Page 59 133 Three quadratic problems......Page 61 134 Iterative solutions of a polynomial equation......Page 62 135 The arithmetic-geometric mean......Page 64 140 Linear difference equations......Page 65 141 Constant coefficients......Page 66 142 Powers of matrices......Page 67 200 Introduction to the Euler methods......Page 72 201 Some numerical experiments......Page 75 202 Calculations with stepsize control......Page 79 203 Calculations with mildly stiff problems......Page 81 204 Calculations with the implicit Euler method......Page 84 210 Formulation of the Euler method......Page 86 212 Global truncation error......Page 87 213 Convergence of the Euler method......Page 89 214 Order of convergence......Page 90 215 Asymptotic error formula......Page 93 216 Stability characteristics......Page 95 217 Local truncation error estimation......Page 100 218 Rounding error......Page 101 220 Introduction......Page 106 221 More computations in a step......Page 107 222 Greater dependence on previous values......Page 108 223 Use of higher derivatives......Page 109 224 Multistep–multistage–multiderivative methods......Page 111 226 Local error estimates......Page 112 231 Second order methods......Page 114 232 The coefficient tableau......Page 115 234 Introduction to order conditions......Page 116 235 Fourth order methods......Page 119 237 Implicit Runge–Kutta methods......Page 120 238 Stability characteristics......Page 121 239 Numerical examples......Page 124 241 Adams methods......Page 126 243 Consistency, stability and convergence......Page 128 244 Predictor–corrector Adams methods......Page 130 245 The Milne device......Page 132 246 Starting methods......Page 133 247 Numerical examples......Page 134 250 Introduction to Taylor series methods......Page 135 251 Manipulation of power series......Page 136 252 An example of a Taylor series solution......Page 137 253 Other methods using higher derivatives......Page 140 254 The use of f derivatives......Page 141 255 Further numerical examples......Page 142 260 Historical introduction......Page 143 261 Pseudo Runge–Kutta methods......Page 144 263 General linear methods......Page 145 264 Numerical examples......Page 148 270 Choice of method......Page 149 271 Variable stepsize......Page 151 272 Interpolation......Page 152 273 Experiments with the Kepler problem......Page 153 274 Experiments with a discontinuous problem......Page 154 300 Rooted trees......Page 158 301 Functions on trees......Page 160 302 Some combinatorial questions......Page 162 304 Enumerating non-rooted trees......Page 165 305 Differentiation......Page 167 306 Taylor’s theorem......Page 169 310 Elementary differentials......Page 171 311 The Taylor expansion of the exact solution......Page 174 312 Elementary weights......Page 176 313 The Taylor expansion of the approximate solution......Page 180 314 Independence of the elementary differentials......Page 181 316 Order conditions for scalar problems......Page 183 317 Independence of elementary weights......Page 184 318 Local truncation error......Page 186 319 Global truncation error......Page 187 320 Methods of orders less than 4......Page 191 321 Simplifying assumptions......Page 192 322 Methods of order 4......Page 196 323 New methods from old......Page 202 324 Order barriers......Page 208 325 Methods of order 5......Page 211 326 Methods of order 6......Page 213 327 Methods of orders greater than 6......Page 216 331 Richardson error estimates......Page 219 332 Methods with built-in estimates......Page 222 333 A class of error-estimating methods......Page 223 334 The methods of Fehlberg......Page 229 335 The methods of Verner......Page 231 336 The methods of Dormand and Prince......Page 232 340 Introduction......Page 234 341 Solvability of implicit equations......Page 235 342 Methods based on Gaussian quadrature......Page 236 343 Reflected methods......Page 240 344 Methods based on Radau and Lobatto quadrature......Page 243 351 Criteria for A-stability......Page 251 352 Padé approximations to the exponential function......Page 253 353 A-stability of Gauss and related methods......Page 259 354 Order stars......Page 261 355 Order arrows and the Ehle barrier......Page 264 356 AN-stability......Page 266 357 Non-linear stability......Page 269 358 BN-stability of collocation methods......Page 273 359 The V and W transformations......Page 275 360 Implementation of implicit Runge–Kutta methods......Page 280 361 Diagonally implicit Runge–Kutta methods......Page 282 362 The importance of high stage order......Page 283 363 Singly implicit methods......Page 287 364 Generalizations of singly implicit methods......Page 292 365 Effective order and DESIRE methods......Page 294 370 Maintaining quadratic invariants......Page 296 371 Examples of symplectic methods......Page 297 372 Order conditions......Page 298 373 Experiments with symplectic methods......Page 299 380 Motivation......Page 301 381 Equivalence classes of Runge–Kutta methods......Page 302 382 The group of Runge–Kutta methods......Page 305 383 The Runge–Kutta group......Page 308 384 A homomorphism between two groups......Page 311 385 A generalization of G_1......Page 312 386 Recursive formula for the product......Page 313 387 Some special elements of G......Page 318 388 Some subgroups and quotient groups......Page 321 389 An algebraic interpretation of effective order......Page 323 391 Optimal sequences......Page 329 392 Acceptance and rejection of steps......Page 331 393 Error per step versus error per unit step......Page 332 394 Control-theoretic considerations......Page 333 395 Solving the implicit equations......Page 334 400 Fundamentals......Page 338 401 Starting methods......Page 339 402 Convergence......Page 340 404 Consistency......Page 341 405 Necessity of conditions for convergence......Page 343 406 Sufficiency of conditions for convergence......Page 345 410 Criteria for order......Page 350 411 Derivation of methods......Page 351 412 Backward difference methods......Page 353 420 Introduction......Page 354 421 Further remarks on error growth......Page 356 422 The underlying one-step method......Page 358 423 Weakly stable methods......Page 360 424 Variable stepsize......Page 361 430 Introduction......Page 363 431 Stability regions......Page 365 432 Examples of the boundary locus method......Page 367 434 Stability of predictor–corrector methods......Page 370 440 Survey of barrier results......Page 373 441 Maximum order for a convergent k-step method......Page 374 442 Order stars for linear multistep methods......Page 377 443 Order arrows for linear multistep methods......Page 379 450 The one-leg counterpart to a linear multistep method......Page 381 451 The concept of G-stability......Page 382 452 Transformations relating one-leg and linear multistep methods......Page 385 454 Concluding remarks on G-stability......Page 386 460 Survey of implementation considerations......Page 387 461 Representation of data......Page 388 462 Variable stepsize for Nordsieck methods......Page 392 463 Local error estimation......Page 393 500 Multivalue–multistage methods......Page 394 501 Transformations of methods......Page 396 502 Runge–Kutta methods as general linear methods......Page 397 503 Linear multistep methods as general linear methods......Page 398 504 Some known unconventional methods......Page 401 505 Some recently discovered general linear methods......Page 403 510 Definitions of consistency and stability......Page 406 511 Covariance of methods......Page 407 512 Definition of convergence......Page 408 513 The necessity of stability......Page 409 514 The necessity of consistency......Page 410 515 Stability and consistency imply convergence......Page 411 520 Introduction......Page 418 521 Methods with maximal stability order......Page 419 522 Outline proof of the Butcher–Chipman conjecture......Page 423 523 Non-linear stability......Page 426 524 Reducible linear multistep methods and G-stability......Page 428 525 G-symplectic methods......Page 429 530 Possible definitions of order......Page 431 531 Local and global truncation errors......Page 433 532 Algebraic analysis of order......Page 434 533 An example of the algebraic approach to order......Page 435 534 The order of a G-symplectic method......Page 437 535 The underlying one-step method......Page 438 541 The types of DIMSIM methods......Page 441 542 Runge–Kutta stability......Page 444 543 Almost Runge–Kutta methods......Page 447 544 Third order, three-stage ARK methods......Page 450 545 Fourth order, four-stage ARK methods......Page 452 546 A fifth order, five-stage method......Page 454 547 ARK methods for stiff problems......Page 455 550 Doubly companion matrices......Page 457 551 Inherent Runge–Kutta stability......Page 459 552 Conditions for zero spectral radius......Page 461 553 Derivation of methods with IRK stability......Page 463 554 Methods with property F......Page 466 555 Some non-stiff methods......Page 467 556 Some stiff methods......Page 468 557 Scale and modify for stability......Page 469 558 Scale and modify for error estimation......Page 471 References......Page 474 Index......Page 480 In recent years the study of numerical methods for solving ordinary differential equations has seen many new developments. This second edition of the author's pioneering text is fully revised and updated to acknowledge many of these developments.  It includes a complete treatment of linear multistep methods whilst maintaining its unique and comprehensive emphasis on Runge-Kutta methods and general linear methods.

Although the specialist topics are taken to an advanced level, the entry point to the volume as a whole is not especially demanding.  Early chapters provide a wide-ranging introduction to differential equations and difference equations together with a survey of numerical differential equation methods, based on the fundamental Euler method with more sophisticated methods presented as generalizations of Euler.

Features of the book include

• Introductory work on differential and difference equations.
• A comprehensive introduction to the theory and practice of solving ordinary differential equations numerically.
• A detailed analysis of Runge-Kutta methods and of linear multistep methods.
• A complete study of general linear methods from both theoretical and practical points of view.
• The latest results on practical general linear methods and their implementation.
• A balance between informal discussion and rigorous mathematical style.
• Examples and exercises integrated into each chapter enhancing the suitability of the book as a course text or a self-study treatise.

Written in a lucid style by one of the worlds leading authorities on numerical methods for ordinary differential equations and drawing upon his vast experience, this new edition provides an accessible and self-contained introduction, ideal for researchers and students following courses on numerical methods, engineering and other sciences.

"Authored by one of the world's leading authorities on numerical methods this update of one of the standard references on numerical analysis, outlines recent developments in the field and presenting a detailed overview of the area. The only book to provide both a detailed treatment of Runge-Kutta methods and a thorough exposition of general linear methods, it also provides practical guidance on solving equations associated with general linear methods, thus providing assistance to those who wish to develop their own computer code. Accompanied by a website hosting solutions to problems and slides for use in teaching Illustrated throughout by worked examples of key algorithms. Presents practical guidance on solving equations associated with general linear methods Gives an introductory overview of the field before going on to describe recent developments. All methods are illustrated with detailed examples and problems sets."--Publisher description