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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 | |
A order star, 51,285.
A® J tensor product, 216, 331.
B relative order star, 59, 67, 287.
B (p) simplifying assumption, 71, 363.
C error constant, 42, 248,262.
C(rj) simplifying assumption, 71, 363.
C+ positive half plane, 52.
C~ negative half plane, 56.
C(/i) companion matrix, 323.
DAT, DAT2 sets of differential algebraic trees, 410, 507.
DATy, DAT2y sets of differential algebraic trees, 410, 507.
DATz, DAT2Z sets of differential algebraic trees, 410, 507.
D(() simplifying assumption, 71.
DA(£) simplifying assumption, 363.
Db(£) simplifying assumption, 363.
di differentiation index, 455.
Dr disc of radius r, 254.
E(y) E -polynomial, 43,96.
F(t) elementary differential, 106,410, 508.
/(£) Fourier transform, 255.
H(p, q) Hamilton function, 543.
I<q (s) Peano kernel, 254.
I\(Z) stability function for y' = \(x)y, 185.
LDAT, LDAT2 sets of differential algebraic trees, 411, 509.
LDATy, LDAT2y sets of differential algebraic trees, 411, 509.
LDATz, LDAT2z sets of differential algebraic trees, 411,509.
t^t) Lagrange polynomial, 499.
L(q, q) Lagrange function, 13,463.
Ls(x) Laguerre polynomial, 96.
LTq set of labelled trees of order q, 106.
P projection, 494.
pD differentiation order, 315.
pi perturbation index, 459.
Pj interpolation order, 315.
Pk(x) (shifted) Legendre polynomial, 78, 202.
Q
projection, 494.
characteristic polynomial, 282, 291.
Padé approximation, 48.
R(z)
stability function, 16, 40,41, 108, 132.
rM)
coefficient of discrete resolvent, 332, 353, 385.
discrete resolvent, 332, 353.
S
stability domain, 16, 241.
fiscal
scaled stability domain, 60.
S(Z)
stability matrix, 353.
sector of A(a) -stability, 250.
5(/I)
stability matrix, 290.
T
kinetic energy, 463, 531.
T
set of trees, 116.
Tm(z)
Chebyshev polynomial, 31.
TW
set of trees for W -methods, 115.
T(vX)
property T, 81.
U
potential energy, 463, 533.
1Mb
norm, 218.
IIMIb
norm in product space, 216, 218.
IIMIIG
norm in product space, 330.
Hb
inner product norm, 307, 356.
«DM1)
coercivity coefficient, 215.
coercivity coefficient, 215.
SD(x)
differentiation error, 314.
local error, 226, 227, 228, 323.
Sj(x)
interpolation error, 314.
sLM(x)
linear multistep error, 322.
SOL(X)
one-leg error, 314.
H(A)
logarithmic norm, 168.
f(C)
multiplier, 343.
V
one-sided Lipschitz constant, 180, 215, 305, 339.
Q
threshold factor, 176.
e{t)
order of a tree, 410, 508.
9(0
generating polynomial, 240.
a(0
generating polynomial, 240.
VB(*)
error growth function, 193.
Vr(x)
error growth function (linear problems), 169.
V
backward difference operator, 242.
A -acceptable approximations, 43.
A -stability
of multistep methods, 241.
of one-step methods, 42f.
of Pade approximations, 58.
of rational approximations, 56f.
of SDIRK methods, 97.
via positive functions, 87.
>1(0)-stable multistep methods, 250.
Ao -stable multistep methods, 251.
A(a)-stability
of BDF methods, 251.
of blended methods, 267.
of Enright methods, 263.
of extrapolation methods, 137, 139.
of modified EBDF methods, 270.
of multistep methods, 250.
of multistep Radau methods, 276.
of RK methods, 45.
of second derivative BDF methods, 265.
A(a) -stable multistep methods of high or-
der, 25 If.
absolutely monotonic functions, 178.
acceleration level, 465.
accuracy barriers for linear multistep meth-
ods, 254f.
Adams methods, 242f, 249, 266.
adjoint differential equation, 462, 467.
algebraic criterion for G -stability, 309.
algebraic stability,
of general linear methods, 356f.
of multivalue methods, 366f.
of RK methods, 181f, 188, 206, 232.
amplifier, 376f, 379.
Andrews' squeezer mechanism, 530f.
A AT-stability,
of RK methods 184f, 200.
of general linear methods, 360.
asymptotic expansions, 135, 428f, 433, 525f.
asymptotic solution
of van der Pol's equation, 372.
automatic stiffness detection, 21.
backward differentiation formulas, see BDF
backward error analysis
for ODEs, 555f.
on manifolds, 559f.
Bader-Deuflhard method, 134f.
Baumgarte stabilization, 470.
B -convergence, 225.
of G -stable one-leg methods, 316.
of multistep methods, 368f.
of order r, 231.
of RK methods, 225f.
of trapezoidal rule, 234.
of variable step sizes, 230.
BDF methods, 2-3, 239, 246, 259, 266, 280,
285, 296, 308,477, 481, 528, 538.
BEAM, 146, 153, 155f, 159, 300, 302.
beam equation, 8f, llf, 20, 38f, 46,146.
BECKDO, 149f, 152, 155f, 300.
Becker-Döring model, 149f.
Bernstein's inequality, 324.
ß -blocked multistep methods, 527.
blended multistep methods, 266.
boundary layer terms, 389.
BRUSS, 148, 155f, 159f, 300, 302.
Brusselator, 6, 19, 31, 148.
BRUSS-2D, 15 If, 157f, 160, 300.
B -stability
of Radau IIA, 199.
of RK methods, 180f, 188, 201.
of Rosenbrock methods, 200.
Burgers equation, 349f, 443f, 448.
Cary Grant's part, 62.
Cash's algorithm, 268.
characteristic equation
for general linear methods, 291.
for linear multistep methods, 240.
for multistep RK methods, 282.
for predictor-corrector schemes, 244.
characterization
of algebraically stable methods, 209.
of positive quadrature formulas, 205.
Chebyshev method, 3If.
of second order, 34f.
Chebyshev polynomial, 3If.
chemical reactions, 3.
Christoffel-Darboux formula, 130.
circuits, 4, 376, 379.
coercivity coefficient 215, 368.
collocation methods
for index 2 DAE, 498.
multi-step, 270f.
one-step, 47, 78.
projected, 503.
singly implicit, 129.
companion matrix, 323.
comparing stability domains, 58.
comparison
between Chebyshev methods, 160.
between extrapolation methods, 159f.
between IRK methods, 158f.
between Radau codes, 158f.
between Rosenbrock codes, 158f.
composite multistep methods, 267.
composition methods 50, 554f.
consistent initial values
for index 1,374, 378.
for index 2,456.
for mechanical systems, 535.
constrained mechanical system, 464,469f,
524, 543.
construction of IRK methods, 83.
continued fraction representation, 50, 85.
continued fractions related to quadrature for-
mulas, 20 If.
continuous solution, see 4dense output'
contractivity
for linear problems, 167f.
in general norms, 175.
see also 4 B -stability'
control problems, 46If.
convergence
for linear problems, 32If.
for nonlinear problems, 339f.
of A-stable multistep methods, 317f.
of BDF for index 2, 486.
of DAE Rosenbrock methods, 416f.
of half-explicit RK methods, 521.
of multistep methods for index 2, 489.
of multistep methods for SPP, 383f.
of RK for index 1,380.
of RK for index 2 DAE, 496f, 504.
of RK methods for DAE, 394f.
of RK methods for SPP, 402.
of symplectic methods, 547, 549.
see also ' B -convergence'
coordinate partitioning, 476,478f.
counter-examples
for existence, 217.
for index definitions, 460f.
for stability properties, 199.
criterion for G -stability, 309.
CUSP, 147, 300, 302.
cusp catastrophe, 147.
DAE, 373, 451.
overdetermined, 477.
Dahlquist's first barrier, 299.
Dahlquist's second barrier, 247,286,297, 299.
Dahlquist's test equation, 16, 240.
damped Chebyshev methods, 32f.
Daniel-Moore conjecture, 51, 286, 294, 298,
364.
DASSL, 481,538, 541.
DEABM, 5, 6.
DEBDF, 301 f.
dense output, 576.
of DAE extrapolation methods, 438f.
of DAE Rosenbrock methods, 422.
of Enright methods, 263f.
r, of multistep collocation methods, 272.
of SDIRK4, 100.
derivative feedback (D), 28.
derivative array equations, 478.
descriptor form, 464.
diagonally implicit RK methods, 9If.
difference-corrected BDF, 528.
differential-algebraic equations, see DAE.
differential equations
linear, 167, 321.
nonlinear, 180, 339.
of singular perturbation type, 37If.
on manifolds, 457, 474f, 544.
perturbed, 556.
quasilinear, 442, 576.
second order, 575.
stiff, 2f.
with invariants, 472f.
differentiation index, 455, 478.
differentiation error, 314.
order, 315, 319.
diffusion, 6.
DIRK, 61, 9If, 208, 221.
disc theorem, 58, 254.
discrete resolvent, 332.
discrete variation of constants formula, 332,
348f.
D J -reducible RK methods, 187.
dominant invariant subspace, 161.
DOPRI5, 3, 19, 22f, 25f, 30, 143, 153f, 469,
471.
for mechanical system, 537.
DOP853, llf, 18, 20, 26, 29.
for mechanical system, 537.
Dormand & Prince methods, 27.
Dorodnicyn's asymptotic formula, 374.
drift-off phenomenon, 468f.
dual order stars, 295.
DUMKA, 34f.
efficiency diagram, 154f, 159f, 301f, 537,53
EKBWH-method, 163f.
elastic beam, 146.
electrical circuits, 4, 376, 379.
elementary differentials, 106.
for index 1 DAE, 410.
for index 2 DAE, 508.
embedded formula for RADAU5, 123.
Enright & Kamel method, 163f.
Enright methods, 261f, 266, 275f.
E -polynomial, 43, 96f.
for Pade approximation, 70.
£-embedding method, 374, 382, 407, 426.
e -expansions for SPP
for exact solution, 388.
for RK solution, 392f.
equivalence
between stability concepts, 186, 188.
of A and B stability, 211.
of A and G -stability, 310f.
error
local, 226, 228f, 405,494.
global 226, 321, 328, 399, 403f.
error bounds for one-leg methods, 314f.
error constant, 247, 286f.
of rational approximations, 42, 52, 61, 67
of second derivative multistep methods, It
for SDBDF methods, 265.
error growth function, 193f, 200, 229.
for linear problems, 169f.
superexponential, 171, 194.
error propagation, 229.
Euler equations, 463.
Euler's method 2,15, 45, 58.
explicit, 2,15, 556.
half-explicit, 519, 525.
implicit, 3,45, 169, 247, 491, 557.
symplectic, 545, 557.
Euler's polyhedral formula, 57.
EULSIM, 140, 160.
existence
of multistep solutions, 306f, 482.
of numerical RK solutions, 215f, 397, 521,
546.
expansion of SPP solutions, 388f.
experiments with multistep codes, 300.
explicit
Adams methods, 242f.
Euler method, 2,15.
Runge-Kutta methods, 16.
midpoint rule, 245, 249.
Nystrom methods, 245.
exponential fitting points, 56.
extended BDF methods, 267.
extended multistep methods, 267f.
extrapolation methods, 18, 131.
for index 1 DAE, 426f.
for quasilinear DAE, 447.
GBS, 18.
E5,145,153f, 300f.
first integral, 472
Fortran codes, 565.
Fourier transform, 148, 255.
fast (FFT), 149, 157.
Gauss methods, 71, 181, 184, 198, 200, 220,
226, 504.
Gaussian quadrature formulas, 202.
Gear & Saad method, 161f.
general linear methods, 290f.
algebraic stability of, 356f.
generalized multistep methods, 261.
generating polynomials, 240.
GGL formulation of mechanical system, 465,
478.
global error, 226.
2. expansion for SPP, 399.
for Prothero & Robinson problem, 328.
of linear multistep methods, 321.
of one-leg methods, 322.
Graeco-Roman transformation, 256.
Green's function, 9.
GRK4A, 110.
Gronwall lemma, 460.
G -stability,
of one-leg methods, 307f.
of BDF2 method, 308, 312.
of general linear methods, 356.
half-explicit methods, 519f.
extrapolation methods, 525.
multistep methods, 527.
Runge-Kutta methods, 520.
Hamiltonian function, 473, 543.
perturbed, 558.
Hamiltonian systems, 472f.
constrained, 543f.
perturbed, 558.
hanging rope, 13f.
HEM5, 538.
Hermite interpolation, 271.
Hessenberg form, 122.
HEX5, 538.
hidden manifold, 454.
high order A(a) -stable multistep methods,
25 If.
high oscillations, 11.
HIHA5, method of Higham & Hall, 26f.
HIRES, 144f, 152f, 159f, 300f.
HLR89, 459
hump, 113,405.
hybrid multistep methods, 267.
hyperbolic problems, 37, 51.
implementation
of extrapolation schemes, 139f.
of IRK methods, 118f.
of Rosenbrock methods, 111.
implicit
Adams methods, 243.
Euler method, 3,45, 169, 247, 491.
midpoint rule, 131, 306.
Milne-Simpson methods, 245, 249.
RK methods, 40f, 71f.
implicit differential equations
Mu' = ip(u), 103, 127, 141, 376, 378f,
408, 426.
M(u)ur=<p(u), 442f, 460, 576.
F(u',u) = 0,452, 459,478.
inconsistent initial values
for DAE Rosenbrock methods, 422f.
index, 452f.
differentiation, 454f.
index 1, 371f, 374, 445, 455, 459, 465,
537.
index 2,456, 458, 460,464, 519, 537.
index 3,456, 458, 464, 537.
of nilpotency, 454.
perturbation, 459.
index reduction, 468f.
inexact Jacobian, 114.
influence of perturbations, 218,484, 493.
integral feedback (I), 28.
interpolation error, 314.
order, 315, 319.
invariants, 472.
IRK(DAE), 376.
irreducible RK methods, 187.
/-stability, 43.
Jeltsch-Nevanlinna theorem, 60, 289.
kinetic energy, 8f, 463, 531.
of mechanical systems, 531, 541.
Kirchhoff's law, 376.
Kreiss matrix theorem, 323.
Kreiss problem, 542.
KS, 148f, 300, 302.
Kuramoto-Sivashinsky equation, 148.
Kuntzmann-Butcher methods, 42f, 71.
labelled trees, 105, 411,509.
LADAMS, 301 f, 304.
Lagrange multipliers, 196f, 464.
Lagrange theory, 8, 13,463.
Lagrange-Hamilton principle, 463.
Laguerre polynomials 96, 129f.
Lebedev's realization, 33.
Legendre polynomials, 71, 78, 202.
LIMEX, 448.
linear problems
contractivity, 167f.
index, 452f, 455.
linearly implicit
Euler method, 138f.
Euler for index 1 DAE, 426f.
Euler for quasilinear DAE, 448.
midpoint rule, 134f, 441.
RK method, 102.
Lipschitz constant, 23.
one-sided, 180.
Lobatto IIIA methods, 42f, 75f, 185, 211,
222, 226, 504.
Lobatto IIIA-IIIB pair, 549f, 563.
Lobatto IIIB methods, 75f, 185, 211, 222,
226.
Lobatto IIIC methods, 75f, 184, 198, 220,
223, 226, 403f, 504.
local coordinates, 475.
local error, 226, 228f, 485, 494.
local state space form, 474.
logarithmic norm 168, 390.
LSODE, 143, 153f, 300f.
LSODI, 481.
L -stability, 44.
of SDIRK methods, 98.
manifold, 457.
matrix pencil, 452, 466.
MEBDF, 303f.
mechanical system, 463, 530f.
METAN1, 140.
metastability, 150.
MEXX for mechanical system, 538.
midpoint rule, 245, 249.
Milne-Simpson methods, 245, 249.
monotonically labelled trees, 105, 411, 509.
Montaigne's ruff, 287.
moving finite elements, 442f.
multibody mechanisms, 530.
multiderivative multistep methods, 282.
multiple real-pole approximations, 67, 98f.
multiplier, 342f.
and nonlinearities, 346.
construction of, 344f.
multistep collocation methods, 270f.
as general linear method, 272.
G -stability of, 361.
multistep methods, 239f.
/3 -blocked, 527.
for index 1, 382f.
for index 2, 481.
for quasilinear DAE, 446f.
of Radau type, 273.
multistep Runge-Kutta methods, 281, 362.
multistep twin, 306.
Navier-Stokes equations, 351.
non-autonomous ODE, 103, 141,408.
nonlinear perturbations, 172.
number of positive weights of QF, 203f.
numerical experiments, 143, 300, 403f, 536f.
numerical work and poles, 283.
Nystrom methods, 245.
ODE, see differential equations.
ODEX, 6,7.
one-leg multistep methods, 305f.
error bounds for, 314.
one-sided Lipschitz condition, 180f, 215, 305,
339, 356.
one-sided Lipschitz constant, 180.
one-step methods, If.
optimal control problems, 461f, 467.
optimal stability regions, 3If.
order conditions
for DAE Rosenbrock methods, 415.
for index 2 DAE, 506f, 512, 523.
for Rosenbrock methods, 104f.
for SDIRK methods, 91f..
for second derivative multistep methods,
261.
order of a tree, 410, 508.
order of B -convergence, 231.
order of a quadrature formula, 202.
order reduction, 225.
for Rosenbrock methods, 236.
order stars, 5If.
dual, 295.
for BDF2, 285.
for general linear methods, 290.
for multistep methods, 279, 284f.
for one-step methods, 51.
for Pade approximations, 53.
for SDIRK methods, 55, 101.
relative, 59, 69, 287.
order tableau
for DAE extrapolation methods, 43If, 441.
OREGO, 144, 152f, 159, 300f.
Oregonator, 13.
overdetermined DAE, 477.
Pade approximations to ez, 48f, 170.
parabolic problems, 3 If, 349f.
Parseval identity, 255, 259.
partitioned Rosenbrock methods, 425.
partitioning methods, 160.
Peano kernel, 254f.
pendulum, 463f, 468, 474.
perturbation index, 459.
perturbations
of linear equations, 348.
of RK solutions, 219, 398.
perturbed asymptotic expansions, 428f, 434,
448.
perturbed differential equation, 556.
perturbed Hamiltonian system, 558.
PHEM56, 538.
PI step size control, 28.
PLATE, 146, 152f, 300f.
plate differential equation, 146.
poles representing numerical work, 283.
position level, 464.
positive functions, 86f, 313.
positive quadrature formulas, 183, 201, 205.
potential energy, 8f, 463, 533.
of mechanical systems, 533, 541.
preconsistency, 359.
predictive controller, 124.
predictor-corrector schemes, 244.
principal root, 285.
principal sheet, 285, 292.
projected collocation methods, 503.
projected Runge-Kutta methods, 502, 515f.
projection methods, 160.
for DAE, 470f.
for ODEs with invariants, 473.
projections (index 2), 487, 494f.
property C,288f.
property T, 81.
proportional feedback (P), 28.
Prothero-Robinson problem, 153, 225, 328,
427.
quasilinear differential equation, 442f, 576.
index 1,445.
Radau IA, 72, 184, 220, 226, 403f, 504.
Radau IIA, 74, 184, 197, 220,226,403f, 504.
Radau methods of multistep type, 273.
RADAUP, 158f, 574.
RADAU5,4f, 46,118f, 143,153f, 379, 566f.
for mechanical system, 539, 541.
rational approximations with real poles, 61.
RATTLE, 548f.
real-pole sandwich, 62.
red-black reduction, 165.
reduced system, 372, 374, 388.
reducible RK methods, 187f.
region of absolute stability, see 'stability do-
main'
region of step-control stability, 26f.
regular matrix pencil, 452, 466.
relative order star, 59, 69, 287.
relative separation, 161.
resolvent (discrete), 332.
Riemann surfaces, 279f.
RKC, 36, 143, 153f.
RKF4(5), 25.
RKF5(4), 24, 26.
ROBER, 144, 152f, 159, 300f.
Robertson reaction, 3, 18, 144.
RODAS, 143, 153f, 158f,420f, 574.
RODAS5, 143, 158f, 422.
root locus curve, 24If.
for BDF methods, 246.
for Enright methods, 263.
for explicit Adams methods, 243.
for implicit Adams methods, 243.
for Milne-Simpson methods, 245.
for Nystrom methods, 245f.
for SDBDF methods, 265.
ROS4, 143.
Rosenbrock methods, 172f.
comparisons, 158f.
contractivity, 172f.
for stiff problems, 102, 102f.
for DAE, 407f, 447.
order reduction, 236.
with inexact Jacobian, 114.
rotation number, 204.
Routh criterion, 89.
Runge-Kutta methods
explicit, 16.
for index 1 problems, 375.
for index 2 DAE, 492f.
for quasilinear DAE, 446f.
for SPP, 392f.
half-explicit, 520.
implicit, 40f, 7If.
projected, 502, 515f.
savings in linear algebra, 540.
scaled stability domain, 60.
Schur's criterion, 278.
SC-stability, 24f.
for Dormand & Prince methods, 27.
SDBDF, 265.
SDIRK code, 128.
SDIRK method, 42, 44, 91, 183, 208, 403,
504.
SDIRK4, 100, 143, 158f.
SECDER, 303f.
second Dahlquist barrier, 247, 254.
second derivative BDF methods, 265.
second derivative multistep methods, 261.
separably stiff problems, 161.
SEULEX, 140, 143, 153f, 160, 575.
SHAKE, 548.
simplified Newton, 119f, 490.
simplifying assumptions, 71, 80f, 183, 206f,
363.
for index 2 DAE, 514.
singly diagonally implicit RK methods, 91.
singly implicit RK methods, 128f.
singular perturbation problems, 37If, 433.
SIRK-methods, 128f.
smoothing step for extrapolation, 133.
SODEX, 140, 143, 160.
SOLOUT, 576.
SPP, see singular perturbation problems.
SPRINT, 30If, 304, 481.
5-reducible RK methods, 188.
stability analysis
forEuler's method, 15.
for explicit RK methods, 16f.
for modified EBDF methods, 269.
for multistep methods, 240f.
for multistep Radau methods, 274f.
for multistep Runge-Kutta methods, 28If.
stability domain, 16.
cross-shaped 39.
of Bader-Deuflhard method, 134.
of BDF methods, 246.
of modified EBDF methods, 270.
of Chebyshev methods, 32f.
of DOPRI methods, 17.
of Enright methods, 263.
of ERK methods, 17.
of explicit Adams methods, 243.
of extrapolated Euler, 139.
of extrapolated trapezoidal rule, 132.
of GBS extrapolation, 19.
of implicit Adams methods, 243.
of implicit Euler method, 246.
of Milne-Simpson methods, 246.
of multistep methods, 240f.
of multistep Radau methods, 276.
of Nystrom methods, 246.
of Pade approximations, 52.
of predictor-corrector schemes, 245.
stability function R(z), 16, 84.
of Chebyshev methods, 32f.
of collocation method, 47.
of DIRK methods, 61.
of DOPRI5,17, 26.
of DOP853, 18.
of extrapolation methods, 132f.
of IRK methods, 40, 84.
of order > s, 47.
of Rosenbrock methods, 108.
of SDIRK methods, 67, 96f.
stability function for y = X(x)y
of IRK methods, 184f.
stability region, see stability domain,
stabilization
Baumgarte, 470.
by projection, 470.
velocity, 47If.
stabilized explicit methods, 3If.
stage order, 226, 369.
starting values for Newton iteration, 120.
state space form, 374f, 474.
state space form method, 375f, 383.
step size selection, 123f.
predictive, 124.
step-control stability, 24f.
stiff, If.
stiff eigenvalues, 161.
stiff eigenvectors, 161.
stiff mechanical system, 541.
stiff stability of multistep methods, 250.
stiff-detest, 144.
stiffly accurate, 227, 552.
RK methods,45, 376.
Rosenbrock methods, 418f.
SDIRK methods, 92f.
stiffness, 2,151.
detection, 21.
stopping criterion, 120.
for Enright & Kamel method, 164.
STRIDE, 129.
Sullivan, Leon, 9.
superconvergence, 500, 554.
superexponential, 171,194.
super-future point, 267.
symplecticity, 544, 547.
symplectic methods, 543f.
Euler, 545, 561.
Lobatto IIIA-IIIB, 550, 563.
second order, 548f, 558, 561f.
tangent space parametrization, 476.
Taylor expansion
for index 2 DAE, 508f.
of DAE Rosenbrock solution, 412f.
of DAE solutions, 411.
of index 2 RK solution, 51 Of.
Taylor series method, 261.
Tchebychef, see Chebyshev.
test problems, 144f.
theorem of von Neumann, 168, 330.
0 -method, 42, 50.
threshold factor, 176,179.
transient phase, 2.
transistor amplifier, 376f, 379.
trapezoidal rule, 45, 131, 185, 234, 247, 306,
357.
trees
for ODE, 92, 105.
for index 1 DAE, 409f.
for index 2 DAE, 507.
for W -methods, 115.
monotonically labelled, 105, 411, 509.
underlying ODE, 455, 478.
uniqueness
of multistep solutions, 306f, 482.
of RK solutions, 219, 397.
van der Houwen & Sommeijer's approach,
35.
van der Pol's equation, 4-5, 144, 372, 403,
406, 566.
Vandermonde matrix, 78.
VDPOL, 144, 153f, 159, 300f.
velocity level, 464.
velocity stabilization, 471.
VODE, 301 f.
Volterra-Lotka model, 556.
von Neumann's theorem, 168, 330.
V -transformation, 78.
W -methods, 114, 136.
weak AN -stability, 360.
weak instability, 245.
Weierstrass-Kronecker form, 452.
work-precision diagram, 154f, 159f, 301 f, 537,
539.
W -transformation, 77f, 183f.
