Numerical methods for ordinary differential equations
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...
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| Main Author | |
|---|---|
| Format | eBook Book |
| Language | English |
| Published |
Chichester
Wiley
2016
John Wiley & Sons, Incorporated Wiley-Blackwell |
| Edition | 3 |
| Subjects | |
| Online Access | Get full text |
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Butcher, J. C. (John Charles) |
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| Notes | Includes bibliographical references (p. [499]-507) and index |
| OCLC | 953662339 |
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| PublicationPlace | Chichester |
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| Publisher | Wiley John Wiley & Sons, Incorporated Wiley-Blackwell |
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| Snippet | A new edition of this classic work, comprehensively revised to present exciting new developments in this important subject The study of numerical methods for... |
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| SubjectTerms | Differential equations Differential equations -- Numerical solutions Numerical solutions |
| TableOfContents | Cover -- Title Page -- Copyright -- Contents -- Foreword -- Preface to the first edition -- Preface to the second edition -- Preface to the third edition -- Chapter 1 Differential and Difference Equations -- 10 Differential Equation Problems -- 100 Introduction to differential equations -- 101 The Kepler problem -- 102 A problem arising from the method of lines -- 103 The simple pendulum -- 104 A chemical kinetics problem -- 105 The Van der Pol equation and limit cycles -- 106 The Lotka-Volterra problem and periodic orbits -- 107 The Euler equations of rigid body rotation -- 11 Differential Equation Theory -- 110 Existence and uniqueness of solutions -- 111 Linear systems of differential equations -- 112 Stiff differential equations -- 12 Further Evolutionary Problems -- 120 Many-body gravitational problems -- 121 Delay problems and discontinuous solutions -- 122 Problems evolving on a sphere -- 123 Further Hamiltonian problems -- 124 Further differential-algebraic problems -- 13 Difference Equation Problems -- 130 Introduction to difference equations -- 131 A linear problem -- 132 The Fibonacci difference equation -- 133 Three quadratic problems -- 134 Iterative solutions of a polynomial equation -- 135 The arithmetic-geometric mean -- 14 Difference Equation Theory -- 140 Linear difference equations -- 141 Constant coefficients -- 142 Powers of matrices -- 15 Location of Polynomial Zeros -- 150 Introduction -- 151 Left half-plane results -- 152 Unit disc results -- Concluding remarks -- Chapter 2 Numerical Differential Equation Methods -- 20 The Euler Method -- 200 Introduction to the Euler method -- 201 Some numerical experiments -- 202 Calculations with stepsize control -- 203 Calculations with mildly stiff problems -- 204 Calculations with the implicit Euler method -- 21 Analysis of the Euler Method -- 210 Formulation of the Euler method 211 Local truncation error -- 212 Global truncation error -- 213 Convergence of the Euler method -- 214 Order of convergence -- 215 Asymptotic error formula -- 216 Stability characteristics -- 217 Local truncation error estimation -- 218 Rounding error -- 22 Generalizations of the Euler Method -- 220 Introduction -- 221 More computations in a step -- 222 Greater dependence on previous values -- 223 Use of higher derivatives -- 224 Multistep-multistage-multiderivative methods -- 225 Implicit methods -- 226 Local error estimates -- 23 Runge-Kutta Methods -- 230 Historical introduction -- 231 Second order methods -- 232 The coefficient tableau -- 233 Third order methods -- 234 Introduction to order conditions -- 235 Fourth order methods -- 236 Higher orders -- 237 Implicit Runge-Kutta methods -- 238 Stability characteristics -- 239 Numerical examples -- 24 Linear MultistepMethods -- 240 Historical introduction -- 241 Adams methods -- 242 General form of linear multistep methods -- 243 Consistency, stability and convergence -- 244 Predictor-corrector Adams methods -- 245 The Milne device -- 246 Starting methods -- 247 Numerical examples -- 25 Taylor Series Methods -- 250 Introduction to Taylor series methods -- 251 Manipulation of power series -- 252 An example of a Taylor series solution -- 253 Other methods using higher derivatives -- 254 The use of f derivatives -- 255 Further numerical examples -- 26 MultivalueMulitistage Methods -- 260 Historical introduction -- 261 Pseudo Runge-Kutta methods -- 262 Two-step Runge-Kutta methods -- 263 Generalized linear multistep methods -- 264 General linear methods -- 265 Numerical examples -- 27 Introduction to Implementation -- 270 Choice of method -- 271 Variable stepsize -- 272 Interpolation -- 273 Experiments with the Kepler problem -- 274 Experiments with a discontinuous problem -- Concluding remarks Chapter 3 Runge-KuttaMethods -- 30 Preliminaries -- 300 Trees and rooted trees -- 301 Trees, forests and notations for trees -- 302 Centrality and centres -- 303 Enumeration of trees and unrooted trees -- 304 Functions on trees -- 305 Some combinatorial questions -- 306 Labelled trees and directed graphs -- 307 Differentiation -- 308 Taylor's theorem -- 31 Order Conditions -- 310 Elementary differentials -- 311 The Taylor expansion of the exact solution -- 312 Elementary weights -- 313 The Taylor expansion of the approximate solution -- 314 Independence of the elementary differentials -- 315 Conditions for order -- 316 Order conditions for scalar problems -- 317 Independence of elementary weights -- 318 Local truncation error -- 319 Global truncation error -- 32 Low Order ExplicitMethods -- 320 Methods of orders less than 4 -- 321 Simplifying assumptions -- 322 Methods of order 4 -- 323 New methods from old -- 324 Order barriers -- 325 Methods of order 5 -- 326 Methods of order 6 -- 327 Methods of order greater than 6 -- 33 Runge-Kutta Methods with Error Estimates -- 330 Introduction -- 331 Richardson error estimates -- 332 Methods with built-in estimates -- 333 A class of error-estimating methods -- 334 The methods of Fehlberg -- 335 The methods of Verner -- 336 The methods of Dormand and Prince -- 34 Implicit Runge-Kutta Methods -- 340 Introduction -- 341 Solvability of implicit equations -- 342 Methods based on Gaussian quadrature -- 343 Reflected methods -- 344 Methods based on Radau and Lobatto quadrature -- 35 Stability of Implicit Runge-Kutta Methods -- 350 A-stability, A(a)-stability and L-stability -- 351 Criteria for A-stability -- 352 Pade approximations to the exponential function -- 353 A-stability of Gauss and related methods -- 354 Order stars -- 355 Order arrows and the Ehle barrier -- 356 AN-stability -- 357 Non-linear stability 530 Possible definitions of order 424 Variable stepsize -- 43 Stability Characteristics -- 430 Introduction -- 431 Stability regions -- 432 Examples of the boundary locus method -- 433 An example of the Schur criterion -- 434 Stability of predictor-corrector methods -- 44 Order and Stability Barriers -- 440 Survey of barrier results -- 441 Maximum order for a convergent k-step method -- 442 Order stars for linear multistep methods -- 443 Order arrows for linear multistep methods -- 45 One-leg Methods and G-stability -- 450 The one-leg counterpart to a linear multistep method -- 451 The concept of G-stability -- 452 Transformations relating one-leg and linear multistep methods -- 453 Effective order interpretation -- 454 Concluding remarks on G-stability -- 46 Implementation Issues -- 460 Survey of implementation considerations -- 461 Representation of data -- 462 Variable stepsize for Nordsieck methods -- 463 Local error estimation -- Concluding remarks -- Chapter 5 General Linear Methods -- 50 RepresentingMethods in General Linear Form -- 500 Multivalue-multistage methods -- 501 Transformations of methods -- 502 Runge-Kutta methods as general linear methods -- 503 Linear multistep methods as general linear methods -- 504 Some known unconventional methods -- 505 Some recently discovered general linear methods -- 51 Consistency, Stability and Convergence -- 510 Definitions of consistency and stability -- 511 Covariance of methods -- 512 Definition of convergence -- 513 The necessity of stability -- 514 The necessity of consistency -- 515 Stability and consistency imply convergence -- 52 The Stability of General Linear Methods -- 520 Introduction -- 521 Methods with maximal stability order -- 522 Outline proof of the Butcher-Chipman conjecture -- 523 Non-linear stability -- 524 Reducible linear multistep methods and G-stability -- 53 The Order of General Linear Methods 358 BN-stability of collocation methods -- 359 The V and W transformations -- 36 Implementable Implicit Runge-Kutta Methods -- 360 Implementation of implicit Runge-Kutta methods -- 361 Diagonally implicit Runge-Kutta methods -- 362 The importance of high stage order -- 363 Singly implicit methods -- 364 Generalizations of singly implicit methods -- 365 Effective order and DESIRE methods -- 37 Implementation Issues -- 370 Introduction -- 371 Optimal sequences -- 372 Acceptance and rejection of steps -- 373 Error per step versus error per unit step -- 374 Control-theoretic considerations -- 375 Solving the implicit equations -- 38 Algebraic Properties of Runge-Kutta Methods -- 380 Motivation -- 381 Equivalence classes of Runge-Kutta methods -- 382 The group of Runge-Kutta tableaux -- 383 The Runge-Kutta group -- 384 A homomorphism between two groups -- 385 A generalization of G1 -- 386 Some special elements of G -- 387 Some subgroups and quotient groups -- 388 An algebraic interpretation of effective order -- 39 Symplectic Runge-Kutta Methods -- 390 Maintaining quadratic invariants -- 391 Hamiltonian mechanics and symplectic maps -- 392 Applications to variational problems -- 393 Examples of symplectic methods -- 394 Order conditions -- 395 Experiments with symplectic methods -- Concluding remarks -- Chapter 4 Linear Multistep Methods -- 40 Preliminaries -- 400 Fundamentals -- 401 Starting methods -- 402 Convergence -- 403 Stability -- 404 Consistency -- 405 Necessity of conditions for convergence -- 406 Sufficiency of conditions for convergence -- 41 The Order of Linear MultistepMethods -- 410 Criteria for order -- 411 Derivation of methods -- 412 Backward difference methods -- 42 Errors and Error Growth -- 420 Introduction -- 421 Further remarks on error growth -- 422 The underlying one-step method -- 423 Weakly stable methods |
| Title | Numerical methods for ordinary differential equations |
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