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| Chapter IV. Stiff Problems - One-Step Methods | ||
| IV.l Examples of Stiff Equations | 2 | |
| Chemical Reaction Systems | 3 | |
| Electrical Circuits | 4 | |
| Diffusion | 6 | |
| A "Stiff' Beam | 8 | |
| High Oscillations | 11 | |
| Exercises | 11 | |
| IV.2 Stability Analysis for Explicit RK Methods | 15 | |
| Stability Analysis for Euler's Method | 15 | |
| Explicit Runge-Kutta Methods | 16 | |
| Extrapolation Methods | 18 | |
| Analysis of the Examples of IV.l | 18 | |
| Automatic Stiffness Detection | 21 | |
| Step-Control Stability | 24 | |
| A PI Step Size Control | 28 | |
| Stabilized Explicit Runge-Kutta Methods | 31 | |
| Exercises | 37 | |
| IV.3 Stability Function of Implicit RK-Methods | 40 | |
| The Stability Function | 40 | |
| A -Stability | 42 | |
| L-Stability and A(a)-Stability | 44 | |
| Numerical Results | 46 | |
| Stability Functions of Order >s | 47 | |
| Pade Approximations to the Exponential Function | 48 | |
| Exercises | 49 | |
| IV.4 Order Stars | 51 | |
| Introduction | 51 | |
| Order and Stability for Rational Approximations | 56 | |
| Stability of Pade Approximations | 58 | |
| Comparing Stability Domains | 58 | |
| Rational Approximations with Real Poles | 61 | |
| The Real-Pole Sandwich | 62 | |
| Multiple Real-Pole Approximations | 67 | |
| Exercises | 70 | |
| IV.5 Construction of Implicit Runge-Kutta Methods | 71 | |
| Gauss Methods | 71 | |
| Radau IA and Radau IIA Methods | 72 | |
| Lobatto IIIA, IIIB and IIIC Methods | 75 | |
| The W -Transformation | 77 | |
| Construction of Implicit Runge-Kutta Methods | 83 | |
| Stability Function | 84 | |
| Positive Functions | 86 | |
| Exercises | 89 | |
| IV.6 Diagonally Implicit RK Methods | 91 | |
| Order Conditions | 91 | |
| Stiffly Accurate SDIRK Methods | 92 | |
| The Stability Function | 96 | |
| Multiple Real-Pole Approximations with R(oo)=0 | 98 | |
| Choice of Method | 99 | |
| Exercises | 100 | |
| IV.7 Rosenbrock-Type Methods | 102 | |
| Derivation of the Method | 102 | |
| Order Conditions | 104 | |
| The Stability Function | 108 | |
| Construction of Methods of Order 4 | 108 | |
| Higher Order Methods ........................................................................Ill | ||
| Implementation of Rosenbrock-Type Methods................................Ill | ||
| The "Hump" | 113 | |
| Methods with Inexact Jacobian (W -Methods) | 114 | |
| Exercises | 117 | |
| IV.8 Implementation of Implicit Runge-Kutta Methods | 118 | |
| Reformulation of the Nonlinear System | 118 | |
| Simplified Newton Iterations | 119 | |
| The Linear System | 121 | |
| Step Size Selection | 123 | |
| Implicit Differential Equations | 127 | |
| AnSDIRK-Code | 128 | |
| SIRK-Methods | 128 | |
| Exercises | 130 | |
| IY.9 Extrapolation Methods | 131 | |
| Extrapolation of Symmetric Methods | 131 | |
| Smoothing | 133 | |
| The Linearly Implicit Mid-Point Rule | 134 | |
| Implicit and Linearly Implicit Euler Method | 138 | |
| Implementation | 139 | |
| Exercises | 142 | |
| IV.10 Numerical Experiments | 143 | |
| TheCodesUsed | 143 | |
| Twelve Test Problems | 144 | |
| Results and Discussion | 152 | |
| Partitioning and Projection Methods | 160 | |
| Exercises | 165 | |
| IV.ll Contractivity for Linear Problems | 167 | |
| Euclidean Norms (Theorem of von Neumann) | 168 | |
| Error Growth Function for Linear Problems | 169 | |
| Small Nonlinear Perturbations | 172 | |
| Contractivity in || - ||oo and || - ||i | 175 | |
| Study of the Threshold Factor | 176 | |
| Absolutely Monotonic Functions | 178 | |
| Exercises | 179 | |
| IY.12 B-Stability and Contractivity | 180 | |
| One-Sided Lipschitz Condition | 180 | |
| B -Stability and Algebraic Stability | 181 | |
| Some Algebraically Stable IRK Methods | 183 | |
| AN -Stability | 184 | |
| Reducible Runge-Kutta Methods | 187 | |
| The Equivalence Theorem for 5-Irreducible Methods | 188 | |
| Error Growth Function | 193 | |
| Computation of <pb(%) | 195 | |
| Exercises | 199 | |
| IV. 13 Positive Quadrature Formulas and B-Stable RK-Methods | 201 | |
| Quadrature Formulas and Related Continued Fractions | 201 | |
| Number of Positive Weights | 203 | |
| Characterization of Positive Quadrature Formulas | 205 | |
| Necessary Conditions for Algebraic Stability | 206 | |
| Characterization of Algebraically Stable Methods | 209 | |
| The "Equivalence" of A-and B-Stability | 211 | |
| Exercises | 213 | |
| IV.14 Existence and Uniqueness of IRK Solutions | 215 | |
| Existence | 215 | |
| A Counterexample | 217 | |
| Influence of Perturbations and Uniqueness | 218 | |
| Computation of a0(A"1) | 220 | |
| Methods with Singular A | 222 | |
| Lobatto mC Methods | 223 | |
| Exercises | 223 | |
| IV. 15 B-Convergence | 225 | |
| The Order Reduction Phenomenon | 225 | |
| The Local Error | 228 | |
| Error Propagation | 229 | |
| B -Convergence for Variable Step Sizes | 230 | |
| B -Convergence Implies Algebraic Stability | 232 | |
| The Trapezoidal Rule | 234 | |
| Order Reduction for Rosenbrock Methods | 236 | |
| Exercises | 237 | |
| Chapter V. Multistep Methods for Stiff Problems | ||
| V.l Stability of Multistep Methods | 240 | |
| The Stability Region | 240 | |
| Adams Methods | 242 | |
| Predictor-Corrector Schemes | 244 | |
| Nystrom Methods | 245 | |
| BDF | 246 | |
| The Second Dahlquist Barrier | 247 | |
| Exercises | 249 | |
| V.2 "Nearly" A-Stable Multistep Methods | 250 | |
| A(cv)-Stability and Stiff Stability | 250 | |
| High Order A(a) -Stable Methods | 251 | |
| Approximating Low Order Methods with High Order Ones | 253 | |
| A Disc Theorem | 254 | |
| Accuracy Barriers for Linear Multistep Methods | 254 | |
| Exercises | 259 | |
| V.3 Generalized Multistep Methods | 261 | |
| Second Derivative Multistep Methods of Enright | 261 | |
| Second Derivative BDF Methods | 265 | |
| Blended Multistep Methods | 266 | |
| Extended Multistep Methods of Cash | 267 | |
| Multistep Collocation Methods | 270 | |
| Methods of "Radau" Type | 273 | |
| Exercises | 275 | |
| Y.4 Order Stars on Riemann Surfaces | 279 | |
| Riemann Surfaces | 279 | |
| Poles Representing Numerical Work | 283 | |
| Order and Order Stars | 284 | |
| The "Daniel and Moore Conjecture" | 286 | |
| Methods with Property C | 288 | |
| General Linear Methods | 290 | |
| Dual Order Stars | 295 | |
| Exercises | 297 | |
| V.5 Experiments with Multistep Codes | 300 | |
| The Codes Used | 300 | |
| Exercises | 304 | |
| V.6 One-Leg Methods and G-Stability | 305 | |
| One-Leg (Multistep) Methods | 305 | |
| Existence and Uniqueness | 306 | |
| G -Stability | 307 | |
| An Algebraic Criterion | 309 | |
| The Equivalence of A-Stability and G -Stability | 310 | |
| A Criterion for Positive Functions | 313 | |
| Error Bounds for One-Leg Methods | 314 | |
| Convergence of A-Stable Multistep Methods | 317 | |
| Exercises | 319 | |
| V.7 Convergence for Linear Problems | 321 | |
| Difference Equations for the Global Error | 321 | |
| The Kreiss Matrix Theorem | 323 | |
| Some Applications of the Kreiss Matrix Theorem | 326 | |
| Global Error for Prothero and Robinson Problem | 328 | |
| Convergence for Linear Systems with Constant Coefficients | 329 | |
| Matrix Valued Theorem of von Neumann | 330 | |
| Discrete Variation of Constants Formula | 332 | |
| Exercises | 337 | |
| V.8 Convergence for Nonlinear Problems | 339 | |
| Problems Satisfying a One-Sided Lipschitz Condition | 339 | |
| Multiplier Technique | 342 | |
| Multipliers and Nonlinearities | 346 | |
| Discrete Variation of Constants and Perturbations | 348 | |
| Convergence for Nonlinear Parabolic Problems | 349 | |
| Exercises | 354 | |
| V.9 Algebraic Stability of General Linear Methods | 356 | |
| G -Stability | 356 | |
| Algebraic Stability | 357 | |
| AN -Stability and Equivalence Results | 359 | |
| Multistep Runge-Kutta Methods | 362 | |
| Simplifying Assumptions | 363 | |
| Quadrature Formulas | 365 | |
| Algebraically Stable Methods of Order 2s | 366 | |
| B -Convergence | 368 | |
| Exercises | 370 | |
| Chapter VI. Singular Perturbation Problems | ||
| and Index 1 Problems | ||
| VI.l Solving Index 1 Problems | 372 | |
| Asymptotic Solution of van der Pol's Equation | 372 | |
| The e -Embedding Method for Problems of Index 1 | 374 | |
| State Space Form Method | 375 | |
| A Transistor Amplifier | 376 | |
| Problems of the Form Mu' = tp(u) | 378 | |
| Convergence of Runge-Kutta Methods | 380 | |
| Exercises | 381 | |
| VI.2 Multistep Methods | 382 | |
| Methods for Index 1 Problems | 382 | |
| Convergence for Singular Perturbation Problems | 383 | |
| Exercises | 387 | |
| VI.3 Epsilon Expansions for Exact and RK Solutions | 388 | |
| Expansion of the Smooth Solution | 388 | |
| Expansions with Boundary Layer Terms | 389 | |
| Estimation of the Remainder | 391 | |
| Expansion of the Runge-Kutta Solution | 392 | |
| Convergence of RK-Methods for Differential-Algebraic Systems | 394 | |
| Existence and Uniqueness of the Runge-Kutta Solution | 397 | |
| Influence of Perturbations | 398 | |
| Estimation of the Remainder in the Numerical Solution | 399 | |
| Numerical Confirmation | 403 | |
| Perturbed Initial Values | 405 | |
| Exercises | 406 | |
| VI.4 RosenbrockMethods | 407 | |
| Definition of the Method | 407 | |
| Derivatives of the Exact Solution | 408 | |
| Trees and Elementary Differentials | 409 | |
| Taylor Expansion of the Exact Solution | 411 | |
| Taylor Expansion of the Numerical Solution | 412 | |
| Order Conditions | 415 | |
| Convergence | 416 | |
| Stiffly Accurate Rosenbrock Methods | 418 | |
| Construction of RODAS, a Stiffly Accurate Embedded Method | 420 | |
| Inconsistent Initial Values | 422 | |
| Exercises | 424 | |
| VI.5 Extrapolation Methods | 426 | |
| Linearly Implicit Euler Discretization | 426 | |
| Perturbed Asymptotic Expansion | 428 | |
| Order Tableau | 431 | |
| Error Expansion for Singular Perturbation Problems | 433 | |
| Dense Output | 438 | |
| Exercises | 441 | |
| VI.6 Quasilinear Problems | 442 | |
| Example: Moving Finite Elements | 442 | |
| Problems of Index One | 445 | |
| Numerical Treatment of C(y)y' = f(y) | 446 | |
| Extrapolation Methods | 447 | |
| Exercises | 448 | |
| Chapter VII. Differential-Algebraic Equations | ||
| of Higher Index | ||
| VII.1 The Index and Various Examples | 452 | |
| Linear Equations with Constant Coefficients | 452 | |
| Differentiation Index | 454 | |
| Differential Equations on Manifolds | 457 | |
| The Perturbation Index | 459 | |
| Control Problems | 461 | |
| Mechanical Systems | 463 | |
| Exercises | 465 | |
| VII.2 Index Reduction Methods | 468 | |
| Index Reduction by Differentiation | 468 | |
| Stabilization by Projection | 470 | |
| Differential Equations with Invariants | 472 | |
| Methods Based on Local State Space Forms | 474 | |
| Overdetermined Differential-Algebraic Equations | 477 | |
| Unstructured Higher Index Problems | 478 | |
| Exercises | 480 | |
| VII.3 Multistep Methods for Index 2 DAE | 481 | |
| Existence and Uniqueness of Numerical Solution | 482 | |
| Influence of Perturbations | 484 | |
| The Local Error | 485 | |
| Convergence for BDF | 486 | |
| General Multistep Methods | 489 | |
| Solution of the Nonlinear System by Simplified Newton | 490 | |
| Exercises | 491 | |
| VII.4 Runge-Kutta Methods for Index 2 DAE | 492 | |
| The Nonlinear System | 492 | |
| Estimation of the Local Error | 494 | |
| Convergence for the y -Component | 496 | |
| Convergence for the 2 -Component | 497 | |
| Collocation Methods | 498 | |
| Superconvergence of Collocation Methods | 500 | |
| Projected Runge-Kutta Methods | 502 | |
| Summary of Convergence Results | 504 | |
| Exercises | 505 | |
| VII.5 Order Conditions for Index 2 DAE | 506 | |
| Derivatives of the Exact Solution | 506 | |
| Trees and Elementary Differentials | 507 | |
| Taylor Expansion of the Exact Solution | 508 | |
| Derivatives of the Numerical Solution | 510 | |
| Order Conditions | 512 | |
| Simplifying Assumptions | 514 | |
| Projected Runge-Kutta Methods | 515 | |
| Exercises | 518 | |
| VII.6 Half-Explicit Methods for Index 2 Systems | 519 | |
| Half-Explicit Runge-Kutta Methods | 520 | |
| Extrapolation Methods | 525 | |
| /3-Blocked Multistep Methods | 527 | |
| Exercises | 529 | |
| VII.7 Computation of Multibody Mechanisms | 530 | |
| Description of the Model | 530 | |
| Fortran Subroutines | 533 | |
| Computation of Consistent Initial Values | 535 | |
| Numerical Computations | 536 | |
| A Stiff Mechanical System | 541 | |
| Exercises | 542 | |
| VII.8 Symplectic Methods for Constrained Hamiltonian Systems | 543 | |
| Properties of the Exact Flow | 544 | |
| First Order Symplectic Method | 545 | |
| SHAKE and RATTLE | 548 | |
| The Lobatto HIA-IIIB Pair | 550 | |
| Composition Methods | 554 | |
| Backward Error Analysis (for ODEs) | 555 | |
| Backward Error Analysis on Manifolds | 559 | |
| Exercises | 562 | |
| Appendix. Fortran Codes | 565 | |
| Driver for the Code RADAU5 | 566 | |
| Subroutine RADAU5 | 568 | |
| Subroutine RADAUP | 574 | |
| Subroutine RODAS | 574 | |
| Subroutine SEULEX | 575 | |
| Problems with Special Structure | 575 | |
| Use of SOLOUT and of Dense Output | 576 | |
| Bibliography | 577 | |
| Symbol Index | 605 | |
| Subject Index | 607 | |
