This book is the most comprehensive, up-to-date account of the popular numerical methods for solving boundary value problems in ordinary differential equations. It aims at a thorough understanding of the field by giving an in-depth analysis of the numerical methods by using decoupling principles. Numerous exercises and real-world examples are used throughout to demonstrate the methods and the theory. Although first published in 1988, this republication remains the most comprehensive theoretical coverage of the subject matter, not available elsewhere in one volume. Many problems, arising in a wide variety of application areas, give rise to mathematical models which form boundary value problems for ordinary differential equations. These problems rarely have a closed form solution, and computer simulation is typically used to obtain their approximate solution. This book discusses methods to carry out such computer simulations in a robust, efficient, and reliable manner.
Author(s): Uri M. Ascher, Robert M. M. Mattheij, Robert D. Russell
Series: Classics in applied mathematics
Publisher: Society for Industrial Mathematics
Year: 1987
Language: English
Pages: 624
CONTENTS ......Page 10
LIST OF EXAMPLES ......Page 20
PREFACE TO THE CLASSICS EDITION ......Page 22
PREFACE ......Page 24
1.1.1 Model problems ......Page 30
1.1.2 General forms for the differential equations ......Page 32
1.1.3 General forms for the boundary conditions ......Page 33
1.2 Boundary Value Problems in Applications ......Page 36
2.1 Errors in Computation ......Page 57
2.2.1 Eigenvalues, transformations, factorizations, projections ......Page 61
2.2.2 Norms, angles, and condition numbers ......Page 65
2.2.3 Gaussian elimination, LUdecomposition ......Page 68
2.2.4 Householder transformations: QUdecomposition ......Page 72
2.2.6 The QR algorithm for eigenvalues ......Page 74
2.2.7 Error analysis ......Page 75
2.3.1 Newton's method ......Page 77
2.3.2 Fixed-point iteration and two basic theorems ......Page 78
2.3.3 Quasi-Newton methods ......Page 86
2.3.4 Quasilinearization ......Page 81
2.4.1 Forms for interpolation polynomials ......Page 84
2.4.2 Osculatory interpolation ......Page 87
2.5.1 Local representations ......Page 88
2.5.2 B-splines ......Page 90
2.5.3 Spline interpolation ......Page 91
2.6.1 Basic rules ......Page 92
2.6.2 Composite formulas ......Page 95
2.7 Initial ValueOrdinary DifferentialEquations ......Page 96
2.7.1 Numerical methods ......Page 97
2.7.2 Consistency, stability and convergence ......Page 100
2.7.3 Stiff problems ......Page 102
2.7.4 Error control and step selection ......Page 103
2.8.1 Norms of functions,spaces, and operators ......Page 106
2.8.2 Roundoff errors and truncation errors ......Page 109
3 THEORY OF ORDINARYDIFFERENTIAL EQUATIONS ......Page 113
3.1.1 Results for boundary value problems ......Page 114
3.1.2 Results for initial value problems ......Page 116
3.2.1 A first-order system ......Page 123
3.2.2 A higher-order ODE ......Page 125
3.3 Stability of Initial Value Problems ......Page 128
3.3.1 Stability and fundamental solutions ......Page 129
3.3.2 The constant coefficient case ......Page 131
3.3.3 The general linear case ......Page 133
3.3.4 The nonlinear case ......Page 138
3.4 Conditioning of Boundary Value Problems ......Page 139
3.4.1 Linear problems and conditioning constants ......Page 140
3.4.2 Dichotomy ......Page 144
3.4.3 Well-conditioning and dichotomy ......Page 150
Exercises ......Page 756
4.1 Introduction: Shooting ......Page 161
4.1.1 Shooting for a linear second-order problem ......Page 162
4.1.3 Shooting—generalapplication, limitations, extensions ......Page 163
4.2.1 Superposition ......Page 164
4.2.2 Numerical accuracy ......Page 166
4.2.3 Numerical stability ......Page 169
4.2.4 Reducedsuperposition ......Page 172
4.3 Multiple Shooting for Linear Problems ......Page 174
4.3.1 The method ......Page 175
4.3.2 Stability ......Page 178
4.4.2 Decoupling ......Page 186
4.3.4 Compactification ......Page 182
4.4 Marching Techniques for Multiple Shooting ......Page 184
4.4.1 Reorthogonalization ......Page 185
4.4.3 Stabilized march ......Page 196
4.5.1 Invariant imbedding ......Page 193
4.5.2 Riccati transformationsfor general linear BVPs ......Page 195
4.5.3 Method properties ......Page 197
4.6.1 Shooting for nonlinear problems ......Page 199
4.6.2 Difficulties with single shooting ......Page 203
4.6.3 Multiple shooting for nonlinear problems ......Page 204
Exercises ......Page 209
5.1 Introduction ......Page 214
5.1.1 A simple schemefor a second-order problem ......Page 216
5.1.2 Simple one-step schemesfor linear systems ......Page 219
5.1.3 Simple schemesfor nonlinear problems ......Page 223
5.2.1 Linear problems ......Page 227
5.2.2 Nonlinear problems ......Page 232
5.3 Higher-Order One-Step Schemes ......Page 237
5.3.1 Implicit Runge-Kutta schemes ......Page 239
5.3.2 A subclass of Runge-Kutta schemes ......Page 242
6.1 Decomposition of Vectors ......Page 306
5.4 Collocation Theory ......Page 247
6.2 Decoupling of the ODE ......Page 308
5.4.2 Nonlinear problems ......Page 251
5.5 Acceleration Techniques ......Page 255
5.5.1 Errorexpansion ......Page 257
5.5.2 Extrapolation ......Page 259
5.5.3 Deferredcorrections ......Page 263
5.5.4 More deferred corrections ......Page 267
5.6 Higher-Order ODEs ......Page 273
5.6.1 More on a simple second-order scheme ......Page 274
5.6.2 Collocation ......Page 276
5.6.3 Collocation implementation using spline bases ......Page 288
5.6.4 Conditioning of collocation matrices ......Page 291
5.7 Finite Element Methods ......Page 295
5.7.1 The Ritz method ......Page 296
Exercises ......Page 301
6 DECOUPLING ......Page 304
6.2.1 Consistent fundamentalsolution ......Page 289
6.2.2 The basic continuous decoupling algorithm ......Page 313
6.3 Decoupling of One-Step Recursions ......Page 317
6.3.1 Consistent fundamental solutions ......Page 319
6.3.2 The basic discretedecoupling algorithm and additional considerations ......Page 320
6.4.1 Consistency for separated BC ......Page 322
6.4.2 Consistencyfor partially separated BC ......Page 324
6.4.3 Consistencyfor general BC ......Page 325
6.5 Closure and its Implications ......Page 326
Exercises ......Page 328
7 SOLVING LINEAR EQUATIONS ......Page 332
7.1 General Staircase Matrices and Condensation ......Page 335
7.2.1 Gaussian elimination with partial pivoting ......Page 337
7.2.2 Alternaterow and column elimination ......Page 339
7.2.3 Block tridiagonal elimination ......Page 342
7.2.4 Stable compactification ......Page 345
7.3 Stability for Block Methods ......Page 347
7.4 Decomposition in the Nonseparated BC Case ......Page 348
7.4.1 The LU-decomposition ......Page 349
7.4.2 A general decoupling algorithm ......Page 350
7.5.1 BVPs with parameters ......Page 351
7.5.2 Multipoint BC ......Page 352
Exercises ......Page 353
8 SOLVING NONLINEAR EQUATIONS ......Page 356
8.1.1 Damped Newton ......Page 358
8.1.2 Altering the Newton direction ......Page 367
8.2.1 Modified Newton ......Page 370
8.3 Finding a Good Initial Guess ......Page 372
8.3.1 Continuation ......Page 373
8.3.2 Imbeddingin atime-dependent problem ......Page 379
8.4 Further Remarks on Discrete Nonlinear BVPS ......Page 382
Exercises ......Page 384
9 MESH SELECTION ......Page 387
9.1 Introduction ......Page 388
9.1.1 Error equidistributionand monitoring ......Page 391
9.2 Direct Methods ......Page 393
9.2.1 Equidistributinglocal truncation error ......Page 394
9.3.1 A practicalmesh selection algorithm ......Page 396
9.3.2 Numerical examples ......Page 400
9.4.1 Explicit method ......Page 402
9.4.2 Implicit method ......Page 404
9.5.1 Somepractical considerations ......Page 409
9.5.2 Coordinating mesh selection and nonlinear iteration ......Page 410
9.5.3 Other approaches ......Page 412
Exercises ......Page 413
10 SINGULAR PERTURBATIONS ......Page 415
10.1 Analytical Approaches ......Page 418
10.1.1 A linear second-order ODE ......Page 419
10.1.2 A nonlinear second-order ODE ......Page 424
10.1.3 Linear first-ordersystems ......Page 426
10.1.4 Nonlinear first-order systems ......Page 440
10.2 Numerical Approaches ......Page 443
10.2.1 Theoretical multiple shooting ......Page 444
10.2.2 Stability andglobal error analysis, I ......Page 448
10.2.3 Stability and global error analysis, II ......Page 451
10.2.4 Thediscretization mesh as a stretching transformation ......Page 457
10.3.1 One-sided schemes ......Page 461
10.3.2 Symmetricschemes ......Page 469
10.3.3 Exponential fitting ......Page 483
10.4.1 Sequential shooting ......Page 486
10.4.2 The Riccati method ......Page 489
Exercises ......Page 496
11.1 Reformulation of Problems in "Standard Form ......Page 498
11.1.1 Higher-order equations,parameters, nonseparated BC, multipoint BC ......Page 499
11.1.2 Conditions at specialpoints ......Page 500
11.1.3 Integralrelations ......Page 502
11.2 Generalized ODEs and Differential Algebraic Equations ......Page 503
11.3.1 Sturm-Liouville problems ......Page 507
11.3.2 Eigenvalues of first-order systems ......Page 511
11.4 BVPswith Singularities; Infinite Intervals ......Page 512
11.4.1 Singularities of the first kind ......Page 513
11.4.2 Infinite interval problems ......Page 515
11.5 Path Following, Singular Points and Bifurcation ......Page 519
11.5.1 Branchingand stability ......Page 522
11.5.2 Numerical techniques ......Page 525
11.6 Highly Oscillatory Solutions ......Page 529
11.7 Functional Differential Equations ......Page 534
11.8 Method of Lines for PDEs ......Page 537
11.9 Multipoint Problems ......Page 541
11.10 On Code Design and Comparison ......Page 544
APPENDIX A: A MULTIPLE SHOOTING CODE ......Page 546
APPENDIX B: A COLLOCATION CODE ......Page 555
REFERENCES ......Page 563
BIBLIOGRAPHY ......Page 595
INDEX ......Page 616
