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Numerical Methods for Ordinary Differential Equations

3. Aufl.

von: J. C. Butcher
89,99 €
Verlag:	Wiley
Format:	EPUB
Veröffentl.:	05.08.2016
ISBN/EAN:	9781119121527
Sprache:	englisch
Anzahl Seiten:	544

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Titelbeschreibung

A new edition of this classic work, comprehensively revised to present exciting new developments in this important subject The study of numerical methods for solving ordinary differential equations is constantly developing and regenerating, and this third edition of a popular classic volume, written by one of the world’s leading experts in the field, presents an account of the subject which reflects both its historical and well-established place in computational science and its vital role as a cornerstone of modern applied mathematics. In addition to serving as a broad and comprehensive study of numerical methods for initial value problems, this book contains a special emphasis on Runge-Kutta methods by the mathematician who transformed the subject into its modern form dating from his classic 1963 and 1972 papers. A second feature is general linear methods which have now matured and grown from being a framework for a unified theory of a wide range of diverse numerical schemes to a source of new and practical algorithms in their own right. As the founder of general linear method research, John Butcher has been a leading contributor to its development; his special role is reflected in the text. The book is written in the lucid style characteristic of the author, and combines enlightening explanations with rigorous and precise analysis. In addition to these anticipated features, the book breaks new ground by including the latest results on the highly efficient G-symplectic methods which compete strongly with the well-known symplectic Runge-Kutta methods for long-term integration of conservative mechanical systems. This third edition of Numerical Methods for Ordinary Differential Equations will serve as a key text for senior undergraduate and graduate courses in numerical analysis, and is an essential resource for research workers in applied mathematics, physics and engineering.

Inhaltsverzeichnis

Foreword xiii Preface to the first edition xv Preface to the second edition xix Preface to the third edition xxi 1 Differential and Difference Equations 1 10 Differential Equation Problems 1 100 Introduction to differential equations 1 101 The Kepler problem 4 102 A problem arising from the method of lines 7 103 The simple pendulum 11 104 A chemical kinetics problem 14 105 The Van der Pol equation and limit cycles 16 106 The Lotka–Volterra problem and periodic orbits 18 107 The Euler equations of rigid body rotation 20 11 Differential Equation Theory 22 110 Existence and uniqueness of solutions 22 111 Linear systems of differential equations 24 112 Stiff differential equations 26 12 Further Evolutionary Problems 28 120 Many-body gravitational problems 28 121 Delay problems and discontinuous solutions 30 122 Problems evolving on a sphere 33 123 Further Hamiltonian problems 35 124 Further differential-algebraic problems 36 13 Difference Equation Problems 38 130 Introduction to difference equations 38 131 A linear problem 39 132 The Fibonacci difference equation 40 133 Three quadratic problems 40 134 Iterative solutions of a polynomial equation 41 135 The arithmetic-geometric mean 43 14 Difference Equation Theory 44 140 Linear difference equations 44 141 Constant coefficients 45 142 Powers of matrices 46 15 Location of Polynomial Zeros 50 150 Introduction 50 151 Left half-plane results 50 152 Unit disc results 52 Concluding remarks 53 2 Numerical Differential Equation Methods 55 20 The Euler Method 55 200 Introduction to the Euler method 55 201 Some numerical experiments 58 202 Calculations with stepsize control 61 203 Calculations with mildly stiff problems 65 204 Calculations with the implicit Euler method 68 21 Analysis of the Euler Method 70 210 Formulation of the Euler method 70 211 Local truncation error 71 212 Global truncation error 72 213 Convergence of the Euler method 73 214 Order of convergence 74 215 Asymptotic error formula 78 216 Stability characteristics 79 217 Local truncation error estimation 84 218 Rounding error 85 22 Generalizations of the Euler Method 90 220 Introduction 90 221 More computations in a step 90 222 Greater dependence on previous values 92 223 Use of higher derivatives 92 224 Multistep–multistage–multiderivative methods 94 225 Implicit methods 95 226 Local error estimates 96 23 Runge–Kutta Methods 97 230 Historical introduction 97 231 Second order methods 98 232 The coefficient tableau 98 233 Third order methods 99 234 Introduction to order conditions 100 235 Fourth order methods 101 236 Higher orders 103 237 Implicit Runge–Kutta methods 103 238 Stability characteristics 104 239 Numerical examples 108 24 Linear MultistepMethods 111 240 Historical introduction 111 241 Adams methods 111 242 General form of linear multistep methods 113 243 Consistency, stability and convergence 113 244 Predictor–corrector Adams methods 115 245 The Milne device 117 246 Starting methods 118 247 Numerical examples 119 25 Taylor Series Methods 120 250 Introduction to Taylor series methods 120 251 Manipulation of power series 121 252 An example of a Taylor series solution 122 253 Other methods using higher derivatives 123 254 The use of f derivatives 126 255 Further numerical examples 126 26 MultivalueMulitistage Methods 128 260 Historical introduction 128 261 Pseudo Runge–Kutta methods 128 262 Two-step Runge–Kutta methods 129 263 Generalized linear multistep methods 130 264 General linear methods 131 265 Numerical examples 133 27 Introduction to Implementation 135 270 Choice of method 135 271 Variable stepsize 136 272 Interpolation 138 273 Experiments with the Kepler problem 138 274 Experiments with a discontinuous problem 139 Concluding remarks 142 3 Runge–KuttaMethods 143 30 Preliminaries 143 300 Trees and rooted trees 143 301 Trees, forests and notations for trees 146 302 Centrality and centres 147 303 Enumeration of trees and unrooted trees 150 304 Functions on trees 153 305 Some combinatorial questions 155 306 Labelled trees and directed graphs 156 307 Differentiation 159 308 Taylor’s theorem 161 31 Order Conditions 163 310 Elementary differentials 163 311 The Taylor expansion of the exact solution 166 312 Elementary weights 168 313 The Taylor expansion of the approximate solution 171 314 Independence of the elementary differentials 174 315 Conditions for order 174 316 Order conditions for scalar problems 175 317 Independence of elementary weights 178 318 Local truncation error 180 319 Global truncation error 181 32 Low Order ExplicitMethods 185 320 Methods of orders less than 4 185 321 Simplifying assumptions 186 322 Methods of order 4 189 323 New methods from old 195 324 Order barriers 200 325 Methods of order 5 204 326 Methods of order 6 206 327 Methods of order greater than 6 209 33 Runge–Kutta Methods with Error Estimates 211 330 Introduction 211 331 Richardson error estimates 211 332 Methods with built-in estimates 214 333 A class of error-estimating methods 215 334 The methods of Fehlberg 221 335 The methods of Verner 223 336 The methods of Dormand and Prince 223 34 Implicit Runge–Kutta Methods 226 340 Introduction 226 341 Solvability of implicit equations 227 342 Methods based on Gaussian quadrature 228 343 Reflected methods 233 344 Methods based on Radau and Lobatto quadrature 236 35 Stability of Implicit Runge–Kutta Methods 243 350 A-stability, A(α)-stability and L-stability 243 351 Criteria for A-stability 244 352 Padé approximations to the exponential function 245 353 A-stability of Gauss and related methods 252 354 Order stars 253 355 Order arrows and the Ehle barrier 256 356 AN-stability 259 357 Non-linear stability 262 358 BN-stability of collocation methods 265 359 The V and W transformations 267 36 Implementable Implicit Runge–Kutta Methods 272 360 Implementation of implicit Runge–Kutta methods 272 361 Diagonally implicit Runge–Kutta methods 273 362 The importance of high stage order 274 363 Singly implicit methods 278 364 Generalizations of singly implicit methods 283 365 Effective order and DESIRE methods 285 37 Implementation Issues 288 370 Introduction 288 371 Optimal sequences 288 372 Acceptance and rejection of steps 290 373 Error per step versus error per unit step 291 374 Control-theoretic considerations 292 375 Solving the implicit equations 293 38 Algebraic Properties of Runge–Kutta Methods 296 380 Motivation 296 381 Equivalence classes of Runge–Kutta methods 297 382 The group of Runge–Kutta tableaux 299 383 The Runge–Kutta group 302 384 A homomorphism between two groups 308 385 A generalization of G1 309 386 Some special elements of G 311 387 Some subgroups and quotient groups 314 388 An algebraic interpretation of effective order 316 39 Symplectic Runge–Kutta Methods 323 390 Maintaining quadratic invariants 323 391 Hamiltonian mechanics and symplectic maps 324 392 Applications to variational problems 325 393 Examples of symplectic methods 326 394 Order conditions 327 395 Experiments with symplectic methods 328 4 Linear Multistep Methods 333 40 Preliminaries 333 400 Fundamentals 333 401 Starting methods 334 402 Convergence 335 403 Stability 336 404 Consistency 336 405 Necessity of conditions for convergence 338 406 Sufficiency of conditions for convergence 339 41 The Order of Linear Multistep Methods 344 410 Criteria for order 344 411 Derivation of methods 346 412 Backward difference methods 347 42 Errors and Error Growth 348 420 Introduction 348 421 Further remarks on error growth 350 422 The underlying one-step method 352 423 Weakly stable methods 354 424 Variable stepsize 355 43 Stability Characteristics 357 430 Introduction 357 431 Stability regions 359 432 Examples of the boundary locus method 360 433 An example of the Schur criterion 363 434 Stability of predictor–corrector methods 364 44 Order and Stability Barriers 367 440 Survey of barrier results 367 441 Maximum order for a convergent k-step method 368 442 Order stars for linear multistep methods 371 443 Order arrows for linear multistep methods 373 45 One-leg Methods and G-stability 375 450 The one-leg counterpart to a linear multistep method 375 451 The concept of G-stability 376 452 Transformations relating one-leg and linear multistep methods 379 453 Effective order interpretation 380 454 Concluding remarks on G-stability 380 46 Implementation Issues 381 460 Survey of implementation considerations 381 461 Representation of data 382 462 Variable stepsize for Nordsieck methods 385 463 Local error estimation 386 Concluding remarks 387 5 General Linear Methods 389 50 RepresentingMethods in General Linear Form 389 500 Multivalue–multistage methods 389 501 Transformations of methods 391 502 Runge–Kutta methods as general linear methods 392 503 Linear multistep methods as general linear methods 393 504 Some known unconventional methods 396 505 Some recently discovered general linear methods 398 51 Consistency, Stability and Convergence 400 510 Definitions of consistency and stability 400 511 Covariance of methods 401 512 Definition of convergence 403 513 The necessity of stability 404 514 The necessity of consistency 404 515 Stability and consistency imply convergence 406 52 The Stability of General Linear Methods 412 520 Introduction 412 521 Methods with maximal stability order 413 522 Outline proof of the Butcher–Chipman conjecture 417 523 Non-linear stability 419 524 Reducible linear multistep methods and G-stability 422 53 The Order of General Linear Methods 423 530 Possible definitions of order 423 531 Local and global truncation errors 425 532 Algebraic analysis of order 426 533 An example of the algebraic approach to order 428 534 The underlying one-step method 429 54 Methods with Runge–Kutta stability 431 540 Design criteria for general linear methods 431 541 The types of DIMSIM methods 432 542 Runge–Kutta stability 435 543 Almost Runge–Kutta methods 438 544 Third order, three-stage ARK methods 441 545 Fourth order, four-stage ARK methods 443 546 A fifth order, five-stage method 446 547 ARK methods for stiff problems 446 55 Methods with Inherent Runge–Kutta Stability 448 550 Doubly companion matrices 448 551 Inherent Runge–Kutta stability 450 552 Conditions for zero spectral radius 452 553 Derivation of methods with IRK stability 454 554 Methods with property F 457 555 Some non-stiff methods 458 556 Some stiff methods 459 557 Scale and modify for stability 460 558 Scale and modify for error estimation 462 56 G-symplectic methods 464 560 Introduction 464 561 The control of parasitism 467 562 Order conditions 471 563 Two fourth order methods 474 564 Starters and finishers for sample methods 476 565 Simulations 480 566 Cohesiveness 481 567 The role of symmetry 487 568 Efficient starting 492 Concluding remarks 497 References 499 Index 509

Autorenportrait

J.C Butcher, Emeritus Professor, University of Auckland, New Zealand

Back cover copy

A new edition of this classic work, comprehensively revised to present exciting new developments in this important subject The study of numerical methods for solving ordinary differential equations is constantly developing and regenerating, and this third edition of a popular classic volume, written by one of the world’s leading experts in the field, presents an account of the subject which reflects both its historical and well-established place in computational science and its vital role as a cornerstone of modern applied mathematics. In addition to serving as a broad and comprehensive study of numerical methods for initial value problems, this book contains a special emphasis on Runge-Kutta methods by the mathematician who transformed the subject into its modern form dating from his classic 1963 and 1972 papers. A second feature is general linear methods which have now matured and grown from being a framework for a unified theory of a wide range of diverse numerical schemes to a source of new and practical algorithms in their own right. As the founder of general linear method research, John Butcher has been a leading contributor to its development; his special role is reflected in the text. The book is written in the lucid style characteristic of the author, and combines enlightening explanations with rigorous and precise analysis. In addition to these anticipated features, the book breaks new ground by including the latest results on the highly efficient G-symplectic methods which compete strongly with the well-known symplectic Runge-Kutta methods for long-term integration of conservative mechanical systems. Key features: ?? Presents a comprehensive and detailed study of the subject ?? Covers both practical and theoretical aspects ?? Includes widely accessible topics along with sophisticated and advanced details ?? Offers a balance between traditional aspects and modern developments This third edition of Numerical Methods for Ordinary Differential Equations will serve as a key text for senior undergraduate and graduate courses in numerical analysis, and is an essential resource for research workers in applied mathematics, physics and engineering.