Numerical Methods for Ordinary Differential Equations.

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.

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.

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.

530 Possible definitions of order.

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