Numerical methods for ordinary differential equations / J.C. Butcher.

Format:

Book

Author/Creator:

Butcher, J. C. (John Charles), 1933-

Contributor:

Rosengarten Family Fund.

Language:

English

Subjects (All):

Differential equations--Numerical solutions.

Differential equations.

Physical Description:

xiv, 425 pages : illustrations ; 24 cm

Place of Publication:

Chichester, West Sussex, England ; Hoboken, NJ : J. Wiley, [2003]

Summary:

In recent years the study of numerical methods for solving ordinary differential equations has seen many new developments. This book is a fully revised update of the author's classic 1987 text, Numerical Analysis of Ordinary Differential Equations, and includes more material on linear multistep methods, whilst maintaining its emphasis on Runge-Kutta methods. It contains introductory material on differential and difference equations, and a comprehensive review of numerical methods and their potential applications. The review starts from the Euler method applied to simple problems and builds on these ideas to introduce increasingly complex methods and problems. The author then explores Runge-Kutta, linear multistep and general linear methods in detail.

Researchers and students from numerical methods, engineering and other sciences will find this book provides an accessible and self-contained introduction to numerical methods for solving ordinary differential equations. It stands out amongst other books on the subject because of the author's lucid writing style, and the integrated presentation of theory, examples, and exercises.

Contents:

1 Differential and Difference Equations 1

10 Differential Equation Problems 1

100 Introduction to differential equations 1

101 The Kepler problem 4

102 Many-body gravitational problems 7

103 A problem arising from the method of lines 9

104 The simple pendulum 13

105 A chemical kinetics problem 15

106 The van der Pol equation and limit cycles 17

107 The Lotka-Volterra problem and periodic orbits 18

11 Differential Equation Theory 23

110 Existence and uniqueness of solutions 23

111 Linear systems of differential equations 25

112 Stiff differential equations 26

113 Laplace transforms 28

12 Difference Equation Problems 31

120 Introduction to difference equations 31

121 A linear problem 31

122 The Fibonacci difference equation 33

123 Three quadratic problems 33

124 Iterative solutions of a polynomial equation 34

125 The arithmetic-geometric mean 36

13 Difference Equation Theory 37

130 Linear difference equations 37

131 Constant coefficients 38

132 Powers of matrices 39

133 The Z-transform 42

2 Numerical Differential Equation Methods 45

20 The Euler Method 45

200 Introduction to the Euler method 45

201 Some numerical experiments 47

202 Calculations with stepsize control 52

203 Calculations with mildly stiff problems 54

204 Calculations with the implicit Euler method 57

21 Analysis of the Euler Method 59

210 Formulation of the Euler method 59

211 Local truncation error 60

212 Global truncation error 61

213 Convergence of the Euler method 62

214 Order of convergence 63

215 Asymptotic error formula 66

216 Stability characteristics 68

217 Local truncation error estimation 72

218 Rounding error 74

22 Generalizations of the Euler Method 78

221 More computations in a step 79

222 Greater dependence on previous values 80

223 Use of higher derivatives 82

224 Multistep-multistage-multiderivative methods 82

225 Implicit methods 83

226 Local error estimates 85

23 Runge-Kutta Methods 86

230 Historical introduction 86

231 Second order methods 86

232 The coefficient tableau 87

233 Third order methods 88

234 Introduction to order conditions 88

235 Fourth order methods 90

236 Higher orders 91

237 Implicit Runge-Kutta methods 92

238 Stability characteristics 93

239 Numerical examples 95

24 Linear Multistep Methods 97

240 Historical introduction 97

241 Adams methods 98

242 General form of linear multistep methods 99

243 Consistency, stability and convergence 100

244 Predictor-corrector Adams methods 102

245 The Milne device 104

246 Starting methods 105

247 Numerical examples 106

25 Taylor Series Methods 107

250 Introduction to Taylor series methods 107

251 Manipulation of power series 108

252 An example of a Taylor series solution 109

253 Other methods using higher derivatives 112

254 The use of f derivatives 113

255 Further numerical examples 114

26 Hybrid Methods 115

260 Historical introduction 115

261 Pseudo Runge-Kutta methods 116

262 Generalized linear multistep methods 117

263 General linear methods 117

264 Numerical examples 120

3 Runge-Kutta Methods 123

300 Rooted trees 123

301 Functions on trees 126

302 Some combinatorial questions 127

303 The use of labelled trees 131

304 Differentiation 131

305 Taylor's theorem 132

31 Order Conditions 134

310 Elementary differentials 134

311 The Taylor expansion of the exact solution 138

312 Elementary weights 140

313 The Taylor expansion of the approximate solution 145

314 Independence of the elementary differentials 146

315 Conditions for order 147

316 Order conditions for scalar problems 148

317 Independence of elementary weights 149

318 Local truncation error 151

319 Global truncation error 152

32 Low Order Explicit Methods 156

320 Methods of orders less than four 156

321 Simplifying assumptions 157

322 Methods of order four 161

323 New methods from old 167

324 Methods of order five 173

325 Methods of order six 175

326 Methods of orders greater than six 178

33 Runge-Kutta Methods with Error Estimates 181

331 Richardson error estimates 181

332 Methods with built-in estimates 184

333 A class of error-estimating methods 185

334 The methods of Fehlberg 191

335 The methods of Verner 193

336 The methods of Dormand and Prince 194

34 Implicit Runge-Kutta Methods 196

341 Solvability of implicit equations 197

342 Methods based on Gaussian quadrature 198

343 Reflected methods 202

344 Methods based on Radau and Lobatto quadrature 205

35 Stability of Implicit Runge-Kutta Methods 212

350 A-stability, A([alpha])-stability and L-stability 212

351 Criteria for A-stability 213

352 Pade approximations to the exponential function 215

353 A-stability of Gauss and related methods 221

354 Order stars 223

355 Order arrows and the Ehle barrier 226

356 AN-stability 228

357 Non-linear stability 232

358 BN-stability of collocation methods 236

359 The V and W transformations 237

36 Implementable Implicit Runge-Kutta Methods 242

360 Implementation of implicit Runge-Kutta methods 242

361 Diagonally-implicit Runge-Kutta methods 244

362 The importance of high stage order 245

363 Singly implicit methods 249

364 Generalizations of singly-implicit methods 254

365 Effective order and DESIRE methods 256

37 Order Barriers 258

370 Explicit barriers 258

371 An upper bound on the required number of stages 260

38 Algebraic Properties of Runge-Kutta Methods 262

380 Motivation 262

381 Equivalence classes of Runge-Kutta methods 263

382 The group of Runge-Kutta methods 266

383 The Runge-Kutta group 269

384 A homomorphism between two groups 272

385 A generalization of G[subscript 1] 274

386 Recursive formula for the product 275

387 Some special elements of G 280

388 Some subgroups and quotient groups 283

389 An algebraic interpretation of effective order 285

39 Implementation Issues 290

391 Optimal sequences 291

392 Acceptance and rejection of steps 293

393 Error per step versus error per unit step 293

394 Control theoretic considerations 295

395 Solving the implicit equations 296

4 Linear Multistep Methods 301

401 Starting methods 302

402 Convergence 303

403 Stability 304

404 Consistency 304

405 Necessity of conditions for convergence 306

406 Sufficiency of conditions for convergence 308

41 The Order of Linear Multistep Methods 312

410 Criteria for order 312

411 Derivation of methods 314

412 Backward difference methods 316

42 Errors and Error Growth 317

421 Further remarks on error growth 319

422 The underlying one-step method 320

423 Weakly stable methods 322

424 Variable stepsize 323

43 Stability Characteristics 326

431 Stability regions 326

432 Examples of the boundary locus method 329

433 Examples of the Schur criterion 332

434 Stability of predictor-corrector methods 333

44 Order and Stability Barriers 335

440 Survey of barrier results 335

441 Maximum order for a convergent k step method 337

442 Order stars for linear multistep methods 339

443 Order arrows for linear multistep methods 341

45 One-leg Methods and G-stability 343

450 The one-leg counterpart to a linear multistep method 343

451 The concept of G-stability 344

452 Transformations relating one-leg and linear multistep methods 347

453 Effective order interpretation 348

454 Concluding remarks on G-stability 348

46 Implementation Issues 349

460 Survey of implementation considerations 349

461 Representation of data 349

462 Variable stepsize for Nordsieck methods 353

463 Local error estimation 354

5 General Linear Methods 357

50 Representing Methods in General Linear Form 357

500 Multivalue multistage methods 357

501 Transformations of methods 358

502 Runge-Kutta methods as general linear methods 360

503 Linear multistep methods as general linear methods 361

504 Some known unconventional methods 364

505 Some recently discovered general linear methods 366

51 Consistency, Stability and Convergence 369

510 Definitions of consistency and stability 369

511 Covariance of methods 370

512 Definition of convergence 371

513 The necessity of stability 372

514 The necessity of consistency 373

515 Stability and consistency imply convergence 374

52 The Stability of General Linear Methods 381

521 Methods with maximal stability order 383

53 The Order of General Linear Methods 387

530 Possible definitions of order 387

531 Algebraic analysis of order 389

532 An example of the algebraic approach to order 390

533 The underlying one-step method 392

54 Methods with Runge-Kutta stability 394

540 Design criteria for general linear methods 394

541 The types of DIMSIM methods 394

542 Runge-Kutta stability 397

543 Almost Runge-Kutta methods 400

544 Fourth order, four stage ARK methods 403

545 Doubly companion matrices 405

546 Inherent Runge-Kutta stability 407

547 Derivation of methods with IRK stability 409

548 Some nonstiff methods 410

549 Some stiff methods 411.

Notes:

Includes bibliographical references (pages [415]-420) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Rosengarten Family Fund.

ISBN:

0471967580

OCLC:

52121487

Online:

Publisher description