Numerical Methods for Ordinary Differential Equations

Preface to the first edition 

Preface to the second edition 

1 Differential and Difference Equations 

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 

135 The arithmetic-geometric mean 

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 Local truncation error 

212 Global truncation error 

213 Convergence of the Euler method 

214 Order of convergence 

215 Asymptotic error formula 

216 Stability characteristics 

217 Local truncation error estimation 

218 Rounding error 

22 Generalizations of the Euler Method 

220 Introduction 

221 More computations in a step 

222 Greater dependence on previous values 

223 Use of higher derivatives 

224 Multistep–multistage–multiderivative methods 

225 Implicit methods 

226 Local error estimates 

23 Runge–Kutta Methods 

230 Historical introduction 

231 Second order methods 

232 The coefficient tableau 

233 Third order methods 

234 Introduction to order conditions 

235 Fourth order methods 

236 Higher orders 

237 Implicit Runge–Kutta methods 

238 Stability characteristics 

239 Numerical examples 

24 Linear Multistep Methods 

240 Historical introduction 

241 Adams methods 

242 General form of linear multistep methods 

243 Consistency, stability and convergence 

244 Predictor–corrector Adams methods 

245 The Milne device 

246 Starting methods 

247 Numerical examples 

25 Taylor Series Methods 

250 Introduction to Taylor series methods 

251 Manipulation of power series 

252 An example of a Taylor series solution 

253 Other methods using higher derivatives 

254 The use of f derivatives 

255 Further numerical examples 

26 Hybrid Methods 

260 Historical introduction 

261 Pseudo Runge–Kutta methods 

262 Generalized linear multistep methods 

263 General linear methods 

264 Numerical examples 

27 Introduction to Implementation 

270 Choice of method 

271 Variable stepsize 

272 Interpolation 

273 Experiments with the Kepler problem 

274 Experiments with a discontinuous problem 

3 Runge–Kutta Methods 

30 Preliminaries 

300 Rooted trees 

301 Functions on trees 

302 Some combinatorial questions 

303 The use of labelled trees 

304 Enumerating non-rooted trees 

305 Differentiation 

306 Taylor’s theorem 

31 Order Conditions 

310 Elementary differentials 

311 The Taylor expansion of the exact solution 

312 Elementary weights 

313 The Taylor expansion of the approximate solution 

314 Independence of the elementary differentials 

315 Conditions for order 

316 Order conditions for scalar problems 

317 Independence of elementary weights 

318 Local 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

327 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 

334 The methods of Fehlberg 

335 The methods of Verner 

336 The methods of Dormand and Prince 

34 Implicit Runge–Kutta Methods 

340 Introduction 

341 Solvability of implicit equations 

342 Methods based on Gaussian quadrature 

343 Reflected methods 

344 Methods based on Radau and Lobatto quadrature 

35 Stability of Implicit Runge–Kutta Methods 

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

351 Criteria for A-stability 

352 Pad´e approximations to the exponential function 

353 A-stability of Gauss and related methods 

354 Order stars 

355 Order arrows and the Ehle barrier 

356 AN-stability 

357 Non-linear stability 

358 BN-stability of collocation methods 

359 The V and W transformations 

36 Implementable Implicit Runge–Kutta Methods 

360 Implementation of implicit Runge–Kutta methods 

361 Diagonally implicit Runge–Kutta methods 

362 The importance of high stage order 

363 Singly implicit methods 

364 Generalizations of singly implicit methods 

365 Effective order and DESIRE methods 

37 Symplectic Runge–Kutta Methods 

370 Maintaining quadratic invariants 

371 Examples of symplectic methods 

372 Order conditions 

373 Experiments with symplectic methods 

38 Algebraic Properties of Runge–Kutta Methods 

380 Motivation 

381 Equivalence classes of Runge–Kutta methods 

382 The group of Runge–Kutta methods 

383 The Runge–Kutta group 

384 A homomorphism between two groups 

385 A generalization of G1

386 Recursive formula for the product 

387 Some special elements of G 

388 Some subgroups and quotient groups 

389 An algebraic interpretation of effective order 

39 Implementation Issues 

390 Introduction 

391 Optimal sequences 

392 Acceptance and rejection of steps 

393 Error per step versus error per unit step 

394 Control-theoretic considerations 

395 Solving the implicit equations 

4 Linear Multistep Methods 

40 Preliminaries 

400 Fundamentals 

401 Starting methods 

402 Convergence 

403 Stability 

404 Consistency 

405 Necessity of conditions for convergence 

406 Sufficiency of conditions for convergence 

41 The Order of Linear Multistep Methods 

410 Criteria for order 

411 Derivation of methods 

412 Backward difference methods 

42 Errors and Error Growth 

420 Introduction 

421 Further remarks on error growth 

422 The underlying one-step method 

423 Weakly stable methods 

424 Variable stepsize 

43 Stability Characteristics 

430 Introduction 

431 Stability regions 

432 Examples of the boundary locus method 

433 An example of the Schur criterion 

434 Stability of predictor–corrector methods 

44 Order and Stability Barriers 

440 Survey of barrier results 

441 Maximum order for a convergent k-step method 

442 Order stars for linear multistep methods 

443 Order arrows for linear multistep methods 

45 One-Leg Methods and G-stability 

450 The one-leg counterpart to a linear multistep method 

451 The concept of G-stability 

452 Transformations relating one-leg and linear multistep methods 

453 Effective order interpretation 

454 Concluding remarks on G-stability 

46 Implementation Issues 

460 Survey of implementation considerations 

461 Representation of data 

462 Variable stepsize for Nordsieck methods 

463 Local error estimation 

5 General Linear Methods 

50 Representing Methods in General Linear Form 

500 Multivalue–multistage methods 

501 Transformations of methods 

502 Runge–Kutta methods as general linear methods 

503 Linear multistep methods as general linear methods 

504 Some known unconventional methods 

505 Some recently discovered general linear methods 

51 Consistency, Stability and Convergence 

510 Definitions of consistency and stability 

511 Covariance of methods 

512 Definition of convergence 

513 The necessity of stability 

514 The necessity of consistency 

515 Stability and consistency imply convergence 

52 The Stability of General Linear Methods 

520 Introduction 

521 Methods with maximal stability order 

522 Outline proof of the Butcher–Chipman conjecture 

523 Non-linear stability 

524 Reducible linear multistep methods and G-stability 

525 G-symplectic methods 

53 The Order of General Linear Methods 

530 Possible definitions of order 

531 Local and global truncation errors 

532 Algebraic analysis of order 

533 An example of the algebraic approach to order 

534 The order of a G-symplectic method 

535 The underlying one-step method 

54 Methods with Runge–Kutta stability 

540 Design criteria for general linear methods 

541 The types of DIMSIM methods 

542 Runge–Kutta stability 

543 Almost Runge–Kutta methods 

544 Third order, three-stage ARK methods 

545 Fourth order, four-stage ARK methods 

546 A fifth order, five-stage method 

547 ARK methods for stiff problems 

55 Methods with Inherent Runge–Kutta Stability 

550 Doubly companion matrices 

551 Inherent Runge–Kutta stability 

552 Conditions for zero spectral radius 

553 Derivation of methods with IRK stability 

554 Methods with property F 

555 Some non-stiff methods 

556 Some stiff methods 

557 Scale and modify for stability 

558 Scale and modify for error estimation 

References 

Index