Scientific Computing with Ordinary Differential Equations

Preface......Page 5
Outline......Page 13
Contents......Page 9
1 Time-Dependent Processes in Science and Engineering......Page 18
1.1 Newtons Celestial Mechanics......Page 21
1.2 Classical Molecular Dynam ics......Page 26
1.3 Chemical Reaction Kinetics......Page 30
1.4 Electrical Circuits......Page 39
Exercises......Page 44
2 Existence and Uniqueness for Initial Value Problems......Page 50
2.1 Global Existence and Uniqueness......Page 51
2.2 Examples of Maximal Continuation......Page 57
2.3 Structure of Nonunique Solutions......Page 61
2.4 Weakly Singular Initial Value Problems......Page 68
2.5 Singular Perturbation Problem s......Page 74
2.6 Quasilinear Differential-Algebraic Problems......Page 77
Exercises......Page 87
3 Condition of Initial Value Problems......Page 92
3.1.1 Propagation Matrices......Page 93
3.1.2 Condition Numbers......Page 99
3.1.3 Perturbation Index of DAE Problems......Page 103
3.2 Stability of ODEs......Page 108
3.2.1 Stability Concept......Page 109
3.2.2 Linear Autonomous ODEs......Page 111
3.2.3 Stability of Fixed P oints......Page 119
3.3.1 Linear Autonomous Recursions......Page 124
3.3.2 Spectra of Rational Matrix Functions......Page 130
Exercises......Page 132
4 One-Step Methods for NonstifF IV Ps......Page 138
4.1 Convergence The o ry......Page 140
4.1.1 Consistency......Page 141
4.1.2 Convergence......Page 142
4.1.3 Concept of S tiffn ess......Page 147
4.2 Explicit Runge-Kutta Methods......Page 150
4.2.1 Concept of Runge-Kutta Methods......Page 151
4.2.2 Classical Runge-Kutta Methods......Page 156
4.2.3 Higher-Order Runge-Kutta Methods......Page 162
4.2.4 Discrete Condition Numbers......Page 171
4.3 Explicit Extrapolation Methods......Page 175
4.3.1 Concept of Extrapolation Methods......Page 176
4.3.2 Asymptotic Expansion of Discretization Error......Page 180
4.3.3 Extrapolation of Explicit Midpoint Rule......Page 185
4.3.4 Extrapolation of Stormer&Verlet Discretization......Page 192
Exercises......Page 200
5 Adaptive Control of One-Step Methods......Page 208
5.1 Local Accuracy Con tro l......Page 210
5.2.1 Excursion to PID C ontrollers......Page 214
5.2.2 Step-size Selection as Controller......Page 217
5.3 Error E stim ation......Page 220
5.4 Embedded Runge-Kutta Methods......Page 224
5.5 Local Versus Achieved Accu racy......Page 230
Exercises......Page 234
6 One-Step Methods for Stiff ODE and DAE IVPs......Page 236
6.1 Inheritance of Asymptotic Stability......Page 239
6.1.1 Rational Approximation of Matrix Exponential......Page 240
6.1.2 Stability Domains......Page 242
6.1.3 Stability Concepts......Page 250
6.1.4 Reversibility and Discrete Isometries......Page 254
6.1.5 Extension to Nonlinear Problem s......Page 257
6.2 Implicit Runge-Kutta Meth ods......Page 261
6.2.1 Stability Functions......Page 268
6.2.2 Solution of Nonlinear Systems......Page 271
6.3 Collocation Methods......Page 275
6.3.1 Basic Idea of Collocation......Page 276
6.3.2 Gauss and Radau Meth ods......Page 284
6.3.3 Dissipative ODEs......Page 288
6.3.4 Conservation of Quadratic First In teg rals......Page 294
6.4.1 Linearly Implicit Runge-Kutta Methods......Page 297
6.4.2 Linearly Implicit Extrapolation Methods......Page 301
6.4.3 Dynamic Elimination of Fast Mod es......Page 310
Exercises......Page 322
7 Multistep Methods for ODE and DAE IV Ps......Page 330
7.1 Multistep Methods on Equidistant Meshes......Page 332
7.1.1 Consistency......Page 336
7.1.2 Stability......Page 340
7.1.3 Convergence......Page 345
7.1.4 Discrete Condition Numbers......Page 354
7.2 Inheritance of Asymptotic Stability......Page 356
7.2.1 Weak Instability in Multistep Methods......Page 358
7.2.2 Linear Stability in Stiff Problem s......Page 361
7.3.1 Adams Methods for Nonstiff ODE Problems......Page 365
7.3.2 BDF Methods for Stiff ODE and DAE Problems......Page 373
7.4 Adaptive Control of Order and Step Size......Page 379
7.4.1 Adams Methods on Variable Meshes......Page 382
7.4.2 BDF Methods on Variable Meshes......Page 384
7.4.3 Nordsieck Representation......Page 393
Exercises......Page 401
8 Boundary Value Problem s for ODEs......Page 406
8.1.1 Local Uniqueness......Page 407
8.1.2 Condition Numbers......Page 410
8.2.1 Shooting Method......Page 414
8.2.2 Multiple Shooting Method......Page 417
8.3 Cyclic Systems of Linear Equations......Page 423
8.3.1 Discrete Condition Numbers......Page 424
8.3.2 Algorithm s......Page 427
8.4 Global Discretization Methods for Spacelike BVPs......Page 432
8.4.1 Elementary Finite Difference Methods......Page 433
8.4.2 Adaptive Collocation Meth ods......Page 440
8.5 More General Types of BVPs......Page 443
8.5.1 Computation of Periodic O rbits......Page 445
8.5.2 Parameter Identification in ODEs......Page 451
8.6 Variational Problem s......Page 457
8.6.1 Classical Variational Problems......Page 458
8.6.2 Optimal Control Problems......Page 466
Exercises......Page 471
References......Page 478
Software......Page 492
Index......Page 494