## Numerical Methods for Ordinary Differential Equations
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.
?œ Presents a comprehensive and detailed study of the subject ?œ Covers both practical and theoretical aspects ?œ Includes widely accessible topics along with sophisticated and advanced details ?œ Offers a balance between traditional aspects and modern developments
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### Contents

Differential and Difference Equations | 1 |

The Kepler problem | 4 |

A problem arising from the method of lines | 7 |

The simple pendulum | 11 |

A chemical kinetics problem | 14 |

The Van der Pol equation and limit cycles | 16 |

The LotkaVolterra problem and periodic orbits | 18 |

The Euler equations of rigid body rotation | 20 |

Order stars | 253 |

Order arrows and the Ehle barrier | 256 |

ANstability | 259 |

Nonlinear stability | 262 |

BNstability of collocation methods | 265 |

The V and W transformations | 267 |

Implementable Implicit RungeKutta Methods | 272 |

Diagonally implicit RungeKutta methods | 273 |

Differential Equation Theory | 22 |

Linear systems of differential equations | 24 |

Stiff differential equations | 26 |

Further Evolutionary Problems | 28 |

Delay problems and discontinuous solutions | 30 |

Problems evolving on a sphere | 33 |

Further Hamiltonian problems | 35 |

Further differentialalgebraic problems | 36 |

Difference Equation Problems | 38 |

A linear problem | 39 |

The Fibonacci difference equation | 40 |

Iterative solutions of a polynomial equation | 41 |

The arithmeticgeometric mean | 43 |

Difference Equation Theory | 44 |

Constant coefficients | 45 |

Powers of matrices | 46 |

Location of Polynomial Zeros | 50 |

Unit disc results | 52 |

Concluding remarks | 53 |

Numerical Differential Equation Methods | 55 |

Some numerical experiments | 58 |

Calculations with stepsize control | 61 |

Calculations with mildly stiff problems | 65 |

Calculations with the implicit Euler method | 68 |

Analysis of the Euler Method | 70 |

Local truncation error | 71 |

Global truncation error | 72 |

Convergence of the Euler method | 73 |

Order of convergence | 74 |

Asymptotic error formula | 78 |

Stability characteristics | 79 |

Local truncation error estimation | 84 |

Rounding error | 85 |

Generalizations of the Euler Method | 90 |

Greater dependence on previous values | 92 |

Multistepmultistagemultiderivative methods | 94 |

Implicit methods | 95 |

Local error estimates | 96 |

RungeKutta Methods | 97 |

Second order methods | 98 |

Third order methods | 99 |

Introduction to order conditions | 100 |

Fourth order methods | 101 |

Higher orders | 103 |

Stability characteristics | 104 |

Numerical examples | 108 |

Linear Multistep Methods | 111 |

General form of linear multistep methods | 113 |

Predictorcorrector Adams methods | 115 |

The Milne device | 117 |

Starting methods | 118 |

Numerical examples | 119 |

Taylor Series Methods | 120 |

Manipulation of power series | 121 |

An example of a Taylor series solution | 122 |

Other methods using higher derivatives | 123 |

The use of f derivatives | 126 |

Multivalue Mulitistage Methods | 128 |

Twostep RungeKutta methods | 129 |

Generalized linear multistep methods | 130 |

General linear methods | 131 |

Numerical examples | 133 |

Introduction to Implementation | 135 |

Variable stepsize | 136 |

Interpolation | 138 |

Experiments with a discontinuous problem | 139 |

Concluding remarks | 142 |

RungeKutta Methods | 143 |

Trees forests and notations for trees | 146 |

Centrality and centres | 147 |

Enumeration of trees and unrooted trees | 150 |

Functions on trees | 153 |

Some combinatorial questions | 155 |

Labelled trees and directed graphs | 156 |

Differentiation | 159 |

Taylors theorem | 161 |

Order Conditions | 163 |

The Taylor expansion of the exact solution | 166 |

Elementary weights | 168 |

The Taylor expansion of the approximate solution | 171 |

Independence of the elementary differentials | 174 |

Order conditions for scalar problems | 175 |

Independence of elementary weights | 178 |

Local truncation error | 180 |

Global truncation error | 181 |

Low Order Explicit Methods | 185 |

Simplifying assumptions | 186 |

Methods of order 4 | 189 |

New methods from old | 195 |

Order barriers | 200 |

Methods of order 5 | 204 |

Methods of order 6 | 206 |

Methods of order greater than 6 | 209 |

RungeKutta Methods with Error Estimates | 211 |

Methods with builtin estimates | 214 |

A class of errorestimating methods | 215 |

The methods of Fehlberg | 221 |

The methods of Verner | 223 |

Implicit RungeKutta Methods | 226 |

Solvability of implicit equations | 227 |

Methods based on Gaussian quadrature | 228 |

Reflected methods | 233 |

Methods based on Radau and Lobatto quadrature | 236 |

Stability of Implicit RungeKutta Methods | 243 |

Criteria for Astability | 244 |

Pade approximations to the exponential function | 246 |

Astability of Gauss and related methods | 252 |

The importance of high stage order | 274 |

Singly implicit methods | 278 |

Generalizations of singly implicit methods | 283 |

Effective order and DESIRE methods | 285 |

Implementation Issues | 288 |

Acceptance and rejection of steps | 290 |

Error per step versus error per unit step | 291 |

Controltheoretic considerations | 292 |

Solving the implicit equations | 293 |

Algebraic Properties of RungeKutta Methods | 296 |

Equivalence classes of RungeKutta methods | 297 |

The group of RungeKutta tableaux | 299 |

The RungeKutta group | 302 |

A homomorphism between two groups | 308 |

A generalization of G1 | 309 |

Some special elements of G | 311 |

Some subgroups and quotient groups | 314 |

An algebraic interpretation of effective order | 316 |

Symplectic RungeKutta Methods | 323 |

Hamiltonian mechanics and symplectic maps | 324 |

Applications to variational problems | 325 |

Examples of symplectic methods | 326 |

Order conditions | 327 |

Experiments with symplectic methods | 328 |

Concluding remarks | 331 |

Linear Multistep Methods | 333 |

Starting methods | 334 |

Convergence | 335 |

Stability | 336 |

Necessity of conditions for convergence | 338 |

Sufficiency of conditions for convergence | 339 |

The Order of Linear Multistep Methods | 344 |

Derivation of methods | 346 |

Backward difference methods | 347 |

Errors and Error Growth | 348 |

Further remarks on error growth | 350 |

The underlying onestep method | 352 |

Weakly stable methods | 354 |

Variable stepsize | 355 |

Stability Characteristics | 357 |

Stability regions | 359 |

Examples of the boundary locus method | 360 |

An example of the Schur criterion | 363 |

Stability of predictorcorrector methods | 364 |

Order and Stability Barriers | 367 |

Maximum order for a convergent kstep method | 368 |

Order stars for linear multistep methods | 371 |

Order arrows for linear multistep methods | 373 |

Oneleg Methods and Gstability | 375 |

The concept of Gstability | 376 |

Transformations relating oneleg and linear multistep methods | 379 |

Effective order interpretation | 380 |

Implementation Issues | 381 |

Representation of data | 382 |

Variable stepsize for Nordsieck methods | 385 |

Local error estimation | 386 |

Concluding remarks | 387 |

General Linear Methods | 389 |

Transformations of methods | 391 |

RungeKutta methods as general linear methods | 392 |

Linear multistep methods as general linear methods | 393 |

Some known unconventional methods | 396 |

Some recently discovered general linear methods | 398 |

Consistency Stability and Convergence | 400 |

Covariance of methods | 401 |

Definition of convergence | 403 |

The necessity of stability | 404 |

Stability and consistency imply convergence | 406 |

The Stability of General Linear Methods | 412 |

Methods with maximal stability order | 413 |

Outline proof of the ButcherChipman conjecture | 417 |

Nonlinear stability | 419 |

Reducible linear multistep methods and Gstability | 422 |

The Order of General Linear Methods | 423 |

Local and global truncation errors | 425 |

Algebraic analysis of order | 426 |

An example of the algebraic approach to order | 428 |

The underlying onestep method | 429 |

Methods with RungeKutta stability | 431 |

The types of DIMSIM methods | 432 |

RungeKutta stability | 435 |

Almost RungeKutta methods | 438 |

Third order threestage ARK methods | 441 |

Fourth order fourstage ARK methods | 443 |

A fifth order fivestage method | 446 |

Methods with Inherent RungeKutta Stability | 448 |

Inherent RungeKutta stability | 450 |

Conditions for zero spectral radius | 452 |

Derivation of methods with IRK stability | 454 |

Methods with property F | 457 |

Some nonstiff methods | 458 |

Some stiff methods | 459 |

Scale and modify for stability | 460 |

Scale and modify for error estimation | 462 |

Gsymplectic methods | 464 |

The control of parasitism | 467 |

Order conditions | 471 |

Two fourth order methods | 474 |

Starters and finishers for sample methods | 476 |

Simulations | 480 |

Cohesiveness | 481 |

The role of symmetry | 487 |

Efficient starting | 492 |

Concluding remarks | 497 |

499 | |

509 | |

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Numerical Methods for Ordinary Differential Equations John Charles Butcher No preview available - 2016 |