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