Numerical Methods for Ordinary Differential Equations / Edition 2 by John C. Butcher | 9780470723357 | Hardcover | Barnes & Noble
Numerical Methods for Ordinary Differential Equations / Edition 2
  • Alternative view 1 of Numerical Methods for Ordinary Differential Equations / Edition 2
  • Alternative view 2 of Numerical Methods for Ordinary Differential Equations / Edition 2

Numerical Methods for Ordinary Differential Equations / Edition 2

by John C. Butcher
     
 

ISBN-10: 0470723351

ISBN-13: 9780470723357

Pub. Date: 05/16/2008

Publisher: Wiley

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

Overview

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.

Product Details

ISBN-13:
9780470723357
Publisher:
Wiley
Publication date:
05/16/2008
Pages:
482
Product dimensions:
6.10(w) x 9.20(h) x 1.30(d)

Table of Contents

Preface to the first edition xi

Preface to the second edition xv

Preface to the third 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 11

104 A chemical kinetics problem 15

105 The Van der Pol equation and limit cycles 17

106 The Lotka–Volterra problem and periodic orbits 18

107 The Euler equations of rigid body rotation 20

11 Differential Equation Theory 23

110 Existence and uniqueness of solutions 23

111 Linear systems of differential equations 25

112 Stiff differential equations 27

12 Further Evolutionary Problems 29

120 Many-body gravitational problems 29

121 Delay problems and discontinuous solutions 32

122 Problems evolving on a sphere 33

123 Further Hamiltonian problems 35

124 Further differential-algebraic problems 37

13 Difference Equation Problems 38

130 Introduction to difference equations 38

131 A linear problem 39

132 The Fibonacci difference equation 40

133 Three quadratic problems 41

134 Iterative solutions of a polynomial equation 42

135 The arithmetic-geometric mean 43

14 Difference Equation Theory 45

140 Linear difference equations 45

141 Constant coefficients 46

142 Powers of matrices 47

15 Location of polynomial zeros 50

150 Introduction 50

151 Left half-plane results 51

152 Unit disc results 53

Concluding remarks 54

2 Numerical Differential Equation Methods 55

20 The Euler Method 55

200 Introduction to the Euler methods 55

201 Some numerical experiments 58

202 Calculations with stepsize control 61

203 Calculations with mildly stiff problems 65

204 Calculations with the implicit Euler method 68

21 Analysis of the Euler Method 70

210 Formulation of the Euler method 70

211 Local truncation error 71

212 Global truncation error 72

213 Convergence of the Euler method 73

214 Order of convergence 74

215 Asymptotic error formula 78

216 Stability characteristics 79

217 Local truncation error estimation 84

218 Rounding error 85

22 Generalizations of the Euler Method 90

220 Introduction 90

221 More computations in a step 90

222 Greater dependence on previous values 92

223 Use of higher derivatives 92

224 Multistep–multistage–multiderivativemethods 94

225 Implicit methods 95

226 Local error estimates 96

23 Runge–KuttaMethods 97

230 Historical introduction 97

231 Second order methods 98

232 The coefficient tableau 98

233 Third order methods 99

234 Introduction to order conditions 100

235 Fourth order methods 102

236 Higher orders 103

237 Implicit Runge–Kutta methods 103

238 Stability characteristics 105

239 Numerical examples 108

24 Linear Multistep Methods 111

240 Historical introduction 111

241 Adams methods 111

242 General form of linear multistep methods 113

243 Consistency, stability and convergence 113

244 Predictor–corrector Adams methods 115

245 The Milne device 117

246 Starting methods 118

247 Numerical examples 119

25 Taylor Series Methods 120

250 Introduction to Taylor series methods 120

251 Manipulation of power series 121

252 An example of a Taylor series solution 122

253 Other methods using higher derivatives 123

254 The use of f derivatives 126

255 Further numerical examples 126

26 Multivalue mulitistage Methods 128

260 Historical introduction 128

261 Pseudo Runge–Kutta methods 128

262 Two-step Runge–Kutta methods 129

263 Generalized linear multistep methods 130

264 General linear methods 131

265 Numerical examples 133

27 Introduction to implementation 135

270 Choice of method 135

271 Variable stepsize 136

272 Interpolation 138

273 Experiments with the Kepler problem 138

274 Experiments with a discontinuous problem 139

Concluding remarks 142

3 Runge–KuttaMethods 143

30 Preliminaries 143

300 Trees and rooted trees 143

301 Trees, forests and notations for trees 146

302 Centrality and centres 147

303 Enumeration of trees and unrooted trees 150

304 Functions on trees 153

305 Some combinatorial questions 155

306 Labelled trees and directed graphs 157

307 Differentiation 158

308 Taylor’s theorem 160

31 Order Conditions 162

310 Elementary differentials 162

311 The Taylor expansion of the exact solution 166

312 Elementary weights 168

313 The Taylor expansion of the approximate solution 172

314 Independence of the elementary differentials 173

315 Conditions for order 174

316 Order conditions for scalar problems 175

317 Independence of elementary weights 178

318 Local truncation error 179

319 Global truncation error 181

32 Low Order Explicit Methods 184

320 Methods of orders less than 4 184

321 Simplifying assumptions 186

322 Methods of order 4 189

323 New methods from old 195

324 Order barriers 201

325 Methods of order 5 204

326 Methods of order 6 206

327 Methods of orders greater than 6 210

33 Runge–KuttaMethods with Error Estimates 212

330 Introduction 212

331 Richardson error estimates 213

332 Methods with built-in estimates 215

333 A class of error-estimating methods 217

334 The methods of Fehlberg 222

335 The methods of Verner 225

336 The methods of Dormand and Prince 225

34 Implicit Runge–KuttaMethods 228

340 Introduction 228

341 Solvability of implicit equations 229

342 Methods based on Gaussian quadrature 230

343 Reflected methods 235

344 Methods based on Radau and Lobatto quadrature 238

35 Stability of Implicit Runge–KuttaMethods 246

350 A-stability, A(α)-stability and L-stability 246

351 Criteria for A-stability 246

352 Padé approximations to the exponential function 248

353 A-stability of Gauss and related methods 254

354 Order stars 256

355 Order arrows and the Ehle barrier 258

356 AN-stability 261

357 Non-linear stability 264

358 BN-stability of collocation methods 268

359 The V and W transformations 270

36 Implementable Implicit Runge–Kutta Methods 274

360 Implementation of implicit Runge–Kutta methods 274

361 Diagonally implicit Runge–Kutta methods 276

362 The importance of high stage order 277

363 Singly implicit methods 281

364 Generalizations of singly implicit methods 286

365 Effective order and DESIRE methods 287

37 Implementation Issues 290

370 Introduction 290

371 Optimal sequences 291

372 Acceptance and rejection of steps 292

373 Error per step versus error per unit step 294

374 Control-theoretic considerations 294

375 Solving the implicit equations 295

38 Algebraic Properties of Runge–Kutta Methods 298

380 Motivation 298

381 Equivalence classes of Runge–Kutta methods 299

382 The group of Runge–Kutta tableaux 302

383 The Runge–Kutta group 306

384 A homomorphism between two groups 311

385 A generalization of G1 313

386 Some special elements of G 314

387 Some subgroups and quotient groups 317

388 An algebraic interpretation of effective order 319

39 Symplectic Runge–Kutta Methods 325

390 Maintaining quadratic invariants 325

391 Hamiltonian mechanics and symplectic maps 326

392 Applications to variational problems 327

393 Examples of symplectic methods 329

394 Order conditions 330

395 Experiments with symplectic methods 331

Concluding remarks 333

4 Linear Multistep Methods 335

40 Preliminaries 335

400 Fundamentals 335

401 Starting methods 336

402 Convergence 337

403 Stability 338

404 Consistency 338

405 Necessity of conditions for convergence 340

406 Sufficiency of conditions for convergence 341

41 The Order of Linear Multistep Methods 346

410 Criteria for order 346

411 Derivation of methods 348

412 Backward difference methods 349

42 Errors and Error Growth 350

420 Introduction 350

421 Further remarks on error growth 352

422 The underlying one-step method 354

423 Weakly stable methods 356

424 Variable stepsize 357

43 Stability Characteristics 359

430 Introduction 359

431 Stability regions 360

432 Examples of the boundary locus method 362

433 An example of the Schur criterion 366

434 Stability of predictor–corrector methods 366

44 Order and Stability Barriers 369

440 Survey of barrier results 369

441 Maximum order for a convergent k-step method 370

442 Order stars for linear multistep methods 373

443 Order arrows for linear multistep methods 375

45 One-Leg Methods and G-stability 377

450 The one-leg counterpart to a linear multistep method 377

451 The concept of G-stability 378

452 Transformations relating one-leg and linear multistep methods381

453 Effective order interpretation 381

454 Concluding remarks on G-stability 382

46 Implementation Issues 383

460 Survey of implementation considerations 383

461 Representation of data 383

462 Variable stepsize for Nordsieck methods 387

463 Local error estimation 388

Concluding remarks 389

5 General Linear Methods 391

50 Representing Methods in General Linear Form 391

500 Multivalue–multistage methods 391

501 Transformations of methods 393

502 Runge–Kutta methods as general linear methods 394

503 Linear multistep methods as general linear methods 395

504 Some known unconventionalmethods 398

505 Some recently discovered general linear methods 400

51 Consistency, Stability and Convergence 403

510 Definitions of consistency and stability 403

511 Covariance of methods 404

512 Definition of convergence 405

513 The necessity of stability 406

514 The necessity of consistency 407

515 Stability and consistency imply convergence 408

52 The Stability of General Linear Methods 415

520 Introduction 415

521 Methods with maximal stability order 416

522 Outline proof of the Butcher–Chipman conjecture 420

523 Non-linear stability 422

524 Reducible linear multistep methods and G-stability 425

53 The Order of General Linear Methods 426

530 Possible definitions of order 426

531 Local and global truncation errors 428

532 Algebraic analysis of order 429

533 An example of the algebraic approach to order 430

534 The underlying one-step method 432

54 Methods with Runge–Kutta stability 434

540 Design criteria for general linear methods 434

541 The types of DIMSIM methods 435

542 Runge–Kutta stability 438

543 Almost Runge–Kutta methods 441

544 Third order, three-stage ARK methods 443

545 Fourth order, four-stage ARK methods 445

546 A fifth order, five-stage method 448

547 ARK methods for stiff problems 449

55 Methods with Inherent Runge–Kutta Stability 450

550 Doubly companion matrices 450

551 Inherent Runge–Kutta stability 453

552 Conditions for zero spectral radius 455

553 Derivation of methods with IRK stability 457

554 Methods with property F 460

555 Some non-stiff methods 461

556 Some stiff methods 461

557 Scale and modify for stability 463

558 Scale and modify for error estimation 465

56 G-symplectic methods 466

560 Introduction 466

561 The control of parasitism 469

562 Order conditions 473

563 Two fourth order methods 476

564 Starters and finishers for sample methods 479

565 Simulations 482

566 Cohesiveness 483

567 The role of symmetry 490

568 Efficient starting 495

Concluding remarks 496

References 499

Index 505

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