{"product_id":"numerical-methods-for-ordinary-differential-equations-isbn-9781119121503","title":"Numerical Methods for Ordinary Differential Equations","description":"\u003cp\u003e\u003cb\u003eA new edition of this classic work, comprehensively revised to present exciting new developments in this important subject\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eThe 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.\u003c\/p\u003e \u003cp\u003eIn 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.\u003c\/p\u003e \u003cp\u003eThis third edition of \u003ci\u003eNumerical Methods for Ordinary Differential Equations\u003c\/i\u003e 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.\u003c\/p\u003e \u003cp\u003e \u003c\/p\u003e \u003cp\u003e \u003c\/p\u003e \u003cp\u003e \u003c\/p\u003e \u003cp\u003eForeword xiii\u003c\/p\u003e \u003cp\u003ePreface to the first edition xv\u003c\/p\u003e \u003cp\u003ePreface to the second edition xix\u003c\/p\u003e \u003cp\u003ePreface to the third edition xxi\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Differential and Difference Equations 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Differential Equation Problems 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e100 Introduction to differential equations 1\u003c\/p\u003e \u003cp\u003e101 The Kepler problem 4\u003c\/p\u003e \u003cp\u003e102 A problem arising from the method of lines 7\u003c\/p\u003e \u003cp\u003e103 The simple pendulum 11\u003c\/p\u003e \u003cp\u003e104 A chemical kinetics problem 14\u003c\/p\u003e \u003cp\u003e105 The Van der Pol equation and limit cycles 16\u003c\/p\u003e \u003cp\u003e106 The Lotka–Volterra problem and periodic orbits 18\u003c\/p\u003e \u003cp\u003e107 The Euler equations of rigid body rotation 20\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Differential Equation Theory 22\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e110 Existence and uniqueness of solutions 22\u003c\/p\u003e \u003cp\u003e111 Linear systems of differential equations 24\u003c\/p\u003e \u003cp\u003e112 Stiff differential equations 26\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Further Evolutionary Problems 28\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e120 Many-body gravitational problems 28\u003c\/p\u003e \u003cp\u003e121 Delay problems and discontinuous solutions 30\u003c\/p\u003e \u003cp\u003e122 Problems evolving on a sphere 33\u003c\/p\u003e \u003cp\u003e123 Further Hamiltonian problems 35\u003c\/p\u003e \u003cp\u003e124 Further differential-algebraic problems 36\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Difference Equation Problems 38\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e130 Introduction to difference equations 38\u003c\/p\u003e \u003cp\u003e131 A linear problem 39\u003c\/p\u003e \u003cp\u003e132 The Fibonacci difference equation 40\u003c\/p\u003e \u003cp\u003e133 Three quadratic problems 40\u003c\/p\u003e \u003cp\u003e134 Iterative solutions of a polynomial equation 41\u003c\/p\u003e \u003cp\u003e135 The arithmetic-geometric mean 43\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Difference Equation Theory 44\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e140 Linear difference equations 44\u003c\/p\u003e \u003cp\u003e141 Constant coefficients 45\u003c\/p\u003e \u003cp\u003e142 Powers of matrices 46\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 Location of Polynomial Zeros 50\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e150 Introduction 50\u003c\/p\u003e \u003cp\u003e151 Left half-plane results 50\u003c\/p\u003e \u003cp\u003e152 Unit disc results 52\u003c\/p\u003e \u003cp\u003eConcluding remarks 53\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Numerical Differential Equation Methods 55\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e20 The Euler Method 55\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e200 Introduction to the Euler method 55\u003c\/p\u003e \u003cp\u003e201 Some numerical experiments 58\u003c\/p\u003e \u003cp\u003e202 Calculations with stepsize control 61\u003c\/p\u003e \u003cp\u003e203 Calculations with mildly stiff problems 65\u003c\/p\u003e \u003cp\u003e204 Calculations with the implicit Euler method 68\u003c\/p\u003e \u003cp\u003e\u003cb\u003e21 Analysis of the Euler Method 70\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e210 Formulation of the Euler method 70\u003c\/p\u003e \u003cp\u003e211 Local truncation error 71\u003c\/p\u003e \u003cp\u003e212 Global truncation error 72\u003c\/p\u003e \u003cp\u003e213 Convergence of the Euler method 73\u003c\/p\u003e \u003cp\u003e214 Order of convergence 74\u003c\/p\u003e \u003cp\u003e215 Asymptotic error formula 78\u003c\/p\u003e \u003cp\u003e216 Stability characteristics 79\u003c\/p\u003e \u003cp\u003e217 Local truncation error estimation 84\u003c\/p\u003e \u003cp\u003e218 Rounding error 85\u003c\/p\u003e \u003cp\u003e\u003cb\u003e22 Generalizations of the Euler Method 90\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e220 Introduction 90\u003c\/p\u003e \u003cp\u003e221 More computations in a step 90\u003c\/p\u003e \u003cp\u003e222 Greater dependence on previous values 92\u003c\/p\u003e \u003cp\u003e223 Use of higher derivatives 92\u003c\/p\u003e \u003cp\u003e224 Multistep–multistage–multiderivative methods 94\u003c\/p\u003e \u003cp\u003e225 Implicit methods 95\u003c\/p\u003e \u003cp\u003e226 Local error estimates 96\u003c\/p\u003e \u003cp\u003e\u003cb\u003e23 Runge–Kutta Methods 97\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e230 Historical introduction 97\u003c\/p\u003e \u003cp\u003e231 Second order methods 98\u003c\/p\u003e \u003cp\u003e232 The coefficient tableau 98\u003c\/p\u003e \u003cp\u003e233 Third order methods 99\u003c\/p\u003e \u003cp\u003e234 Introduction to order conditions 100\u003c\/p\u003e \u003cp\u003e235 Fourth order methods 101\u003c\/p\u003e \u003cp\u003e236 Higher orders 103\u003c\/p\u003e \u003cp\u003e237 Implicit Runge–Kutta methods 103\u003c\/p\u003e \u003cp\u003e238 Stability characteristics 104\u003c\/p\u003e \u003cp\u003e239 Numerical examples 108\u003c\/p\u003e \u003cp\u003e\u003cb\u003e24 Linear MultistepMethods 111\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e240 Historical introduction 111\u003c\/p\u003e \u003cp\u003e241 Adams methods 111\u003c\/p\u003e \u003cp\u003e242 General form of linear multistep methods 113\u003c\/p\u003e \u003cp\u003e243 Consistency, stability and convergence 113\u003c\/p\u003e \u003cp\u003e244 Predictor–corrector Adams methods 115\u003c\/p\u003e \u003cp\u003e245 The Milne device 117\u003c\/p\u003e \u003cp\u003e246 Starting methods 118\u003c\/p\u003e \u003cp\u003e247 Numerical examples 119\u003c\/p\u003e \u003cp\u003e\u003cb\u003e25 Taylor Series Methods 120\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e250 Introduction to Taylor series methods 120\u003c\/p\u003e \u003cp\u003e251 Manipulation of power series 121\u003c\/p\u003e \u003cp\u003e252 An example of a Taylor series solution 122\u003c\/p\u003e \u003cp\u003e253 Other methods using higher derivatives 123\u003c\/p\u003e \u003cp\u003e254 The use of f derivatives 126\u003c\/p\u003e \u003cp\u003e255 Further numerical examples 126\u003c\/p\u003e \u003cp\u003e\u003cb\u003e26 MultivalueMulitistage Methods 128\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e260 Historical introduction 128\u003c\/p\u003e \u003cp\u003e261 Pseudo Runge–Kutta methods 128\u003c\/p\u003e \u003cp\u003e262 Two-step Runge–Kutta methods 129\u003c\/p\u003e \u003cp\u003e263 Generalized linear multistep methods 130\u003c\/p\u003e \u003cp\u003e264 General linear methods 131\u003c\/p\u003e \u003cp\u003e265 Numerical examples 133\u003c\/p\u003e \u003cp\u003e\u003cb\u003e27 Introduction to Implementation 135\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e270 Choice of method 135\u003c\/p\u003e \u003cp\u003e271 Variable stepsize 136\u003c\/p\u003e \u003cp\u003e272 Interpolation 138\u003c\/p\u003e \u003cp\u003e273 Experiments with the Kepler problem 138\u003c\/p\u003e \u003cp\u003e274 Experiments with a discontinuous problem 139\u003c\/p\u003e \u003cp\u003eConcluding remarks 142\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Runge–KuttaMethods 143\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e30 Preliminaries 143\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e300 Trees and rooted trees 143\u003c\/p\u003e \u003cp\u003e301 Trees, forests and notations for trees 146\u003c\/p\u003e \u003cp\u003e302 Centrality and centres 147\u003c\/p\u003e \u003cp\u003e303 Enumeration of trees and unrooted trees 150\u003c\/p\u003e \u003cp\u003e304 Functions on trees 153\u003c\/p\u003e \u003cp\u003e305 Some combinatorial questions 155\u003c\/p\u003e \u003cp\u003e306 Labelled trees and directed graphs 156\u003c\/p\u003e \u003cp\u003e307 Differentiation 159\u003c\/p\u003e \u003cp\u003e308 Taylor’s theorem 161\u003c\/p\u003e \u003cp\u003e\u003cb\u003e31 Order Conditions 163\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e310 Elementary differentials 163\u003c\/p\u003e \u003cp\u003e311 The Taylor expansion of the exact solution 166\u003c\/p\u003e \u003cp\u003e312 Elementary weights 168\u003c\/p\u003e \u003cp\u003e313 The Taylor expansion of the approximate solution 171\u003c\/p\u003e \u003cp\u003e314 Independence of the elementary differentials 174\u003c\/p\u003e \u003cp\u003e315 Conditions for order 174\u003c\/p\u003e \u003cp\u003e316 Order conditions for scalar problems 175\u003c\/p\u003e \u003cp\u003e317 Independence of elementary weights 178\u003c\/p\u003e \u003cp\u003e318 Local truncation error 180\u003c\/p\u003e \u003cp\u003e319 Global truncation error 181\u003c\/p\u003e \u003cp\u003e\u003cb\u003e32 Low Order ExplicitMethods 185\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e320 Methods of orders less than 4 185\u003c\/p\u003e \u003cp\u003e321 Simplifying assumptions 186\u003c\/p\u003e \u003cp\u003e322 Methods of order 4 189\u003c\/p\u003e \u003cp\u003e323 New methods from old 195\u003c\/p\u003e \u003cp\u003e324 Order barriers 200\u003c\/p\u003e \u003cp\u003e325 Methods of order 5 204\u003c\/p\u003e \u003cp\u003e326 Methods of order 6 206\u003c\/p\u003e \u003cp\u003e327 Methods of order greater than 6 209\u003c\/p\u003e \u003cp\u003e\u003cb\u003e33 Runge–Kutta Methods with Error Estimates 211\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e330 Introduction 211\u003c\/p\u003e \u003cp\u003e331 Richardson error estimates 211\u003c\/p\u003e \u003cp\u003e332 Methods with built-in estimates 214\u003c\/p\u003e \u003cp\u003e333 A class of error-estimating methods 215\u003c\/p\u003e \u003cp\u003e334 The methods of Fehlberg 221\u003c\/p\u003e \u003cp\u003e335 The methods of Verner 223\u003c\/p\u003e \u003cp\u003e336 The methods of Dormand and Prince 223\u003c\/p\u003e \u003cp\u003e\u003cb\u003e34 Implicit Runge–Kutta Methods 226\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e340 Introduction 226\u003c\/p\u003e \u003cp\u003e341 Solvability of implicit equations 227\u003c\/p\u003e \u003cp\u003e342 Methods based on Gaussian quadrature 228\u003c\/p\u003e \u003cp\u003e343 Reflected methods 233\u003c\/p\u003e \u003cp\u003e344 Methods based on Radau and Lobatto quadrature 236\u003c\/p\u003e \u003cp\u003e\u003cb\u003e35 Stability of Implicit Runge–Kutta Methods 243\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e350 A-stability, A(α)-stability and L-stability 243\u003c\/p\u003e \u003cp\u003e351 Criteria for A-stability 244\u003c\/p\u003e \u003cp\u003e352 Padé approximations to the exponential function 245\u003c\/p\u003e \u003cp\u003e353 A-stability of Gauss and related methods 252\u003c\/p\u003e \u003cp\u003e354 Order stars 253\u003c\/p\u003e \u003cp\u003e355 Order arrows and the Ehle barrier 256\u003c\/p\u003e \u003cp\u003e356 AN-stability 259\u003c\/p\u003e \u003cp\u003e357 Non-linear stability 262\u003c\/p\u003e \u003cp\u003e358 BN-stability of collocation methods 265\u003c\/p\u003e \u003cp\u003e359 The V and W transformations 267\u003c\/p\u003e \u003cp\u003e\u003cb\u003e36 Implementable Implicit Runge–Kutta Methods 272\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e360 Implementation of implicit Runge–Kutta methods 272\u003c\/p\u003e \u003cp\u003e361 Diagonally implicit Runge–Kutta methods 273\u003c\/p\u003e \u003cp\u003e362 The importance of high stage order 274\u003c\/p\u003e \u003cp\u003e363 Singly implicit methods 278\u003c\/p\u003e \u003cp\u003e364 Generalizations of singly implicit methods 283\u003c\/p\u003e \u003cp\u003e365 Effective order and DESIRE methods 285\u003c\/p\u003e \u003cp\u003e\u003cb\u003e37 Implementation Issues 288\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e370 Introduction 288\u003c\/p\u003e \u003cp\u003e371 Optimal sequences 288\u003c\/p\u003e \u003cp\u003e372 Acceptance and rejection of steps 290\u003c\/p\u003e \u003cp\u003e373 Error per step versus error per unit step 291\u003c\/p\u003e \u003cp\u003e374 Control-theoretic considerations 292\u003c\/p\u003e \u003cp\u003e375 Solving the implicit equations 293\u003c\/p\u003e \u003cp\u003e\u003cb\u003e38 Algebraic Properties of Runge–Kutta Methods 296\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e380 Motivation 296\u003c\/p\u003e \u003cp\u003e381 Equivalence classes of Runge–Kutta methods 297\u003c\/p\u003e \u003cp\u003e382 The group of Runge–Kutta tableaux 299\u003c\/p\u003e \u003cp\u003e383 The Runge–Kutta group 302\u003c\/p\u003e \u003cp\u003e384 A homomorphism between two groups 308\u003c\/p\u003e \u003cp\u003e385 A generalization of G1 309\u003c\/p\u003e \u003cp\u003e386 Some special elements of G 311\u003c\/p\u003e \u003cp\u003e387 Some subgroups and quotient groups 314\u003c\/p\u003e \u003cp\u003e388 An algebraic interpretation of effective order 316\u003c\/p\u003e \u003cp\u003e\u003cb\u003e39 Symplectic Runge–Kutta Methods 323\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e390 Maintaining quadratic invariants 323\u003c\/p\u003e \u003cp\u003e391 Hamiltonian mechanics and symplectic maps 324\u003c\/p\u003e \u003cp\u003e392 Applications to variational problems 325\u003c\/p\u003e \u003cp\u003e393 Examples of symplectic methods 326\u003c\/p\u003e \u003cp\u003e394 Order conditions 327\u003c\/p\u003e \u003cp\u003e395 Experiments with symplectic methods 328\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Linear Multistep Methods 333\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e40 Preliminaries 333\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e400 Fundamentals 333\u003c\/p\u003e \u003cp\u003e401 Starting methods 334\u003c\/p\u003e \u003cp\u003e402 Convergence 335\u003c\/p\u003e \u003cp\u003e403 Stability 336\u003c\/p\u003e \u003cp\u003e404 Consistency 336\u003c\/p\u003e \u003cp\u003e405 Necessity of conditions for convergence 338\u003c\/p\u003e \u003cp\u003e406 Sufficiency of conditions for convergence 339\u003c\/p\u003e \u003cp\u003e\u003cb\u003e41 The Order of Linear Multistep Methods 344\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e410 Criteria for order 344\u003c\/p\u003e \u003cp\u003e411 Derivation of methods 346\u003c\/p\u003e \u003cp\u003e412 Backward difference methods 347\u003c\/p\u003e \u003cp\u003e\u003cb\u003e42 Errors and Error Growth 348\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e420 Introduction 348\u003c\/p\u003e \u003cp\u003e421 Further remarks on error growth 350\u003c\/p\u003e \u003cp\u003e422 The underlying one-step method 352\u003c\/p\u003e \u003cp\u003e423 Weakly stable methods 354\u003c\/p\u003e \u003cp\u003e424 Variable stepsize 355\u003c\/p\u003e \u003cp\u003e\u003cb\u003e43 Stability Characteristics 357\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e430 Introduction 357\u003c\/p\u003e \u003cp\u003e431 Stability regions 359\u003c\/p\u003e \u003cp\u003e432 Examples of the boundary locus method 360\u003c\/p\u003e \u003cp\u003e433 An example of the Schur criterion 363\u003c\/p\u003e \u003cp\u003e434 Stability of predictor–corrector methods 364\u003c\/p\u003e \u003cp\u003e\u003cb\u003e44 Order and Stability Barriers 367\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e440 Survey of barrier results 367\u003c\/p\u003e \u003cp\u003e441 Maximum order for a convergent k-step method 368\u003c\/p\u003e \u003cp\u003e442 Order stars for linear multistep methods 371\u003c\/p\u003e \u003cp\u003e443 Order arrows for linear multistep methods 373\u003c\/p\u003e \u003cp\u003e\u003cb\u003e45 One-leg Methods and G-stability 375\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e450 The one-leg counterpart to a linear multistep method 375\u003c\/p\u003e \u003cp\u003e451 The concept of G-stability 376\u003c\/p\u003e \u003cp\u003e452 Transformations relating one-leg and linear multistep methods 379\u003c\/p\u003e \u003cp\u003e453 Effective order interpretation 380\u003c\/p\u003e \u003cp\u003e454 Concluding remarks on G-stability 380\u003c\/p\u003e \u003cp\u003e\u003cb\u003e46 Implementation Issues 381\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e460 Survey of implementation considerations 381\u003c\/p\u003e \u003cp\u003e461 Representation of data 382\u003c\/p\u003e \u003cp\u003e462 Variable stepsize for Nordsieck methods 385\u003c\/p\u003e \u003cp\u003e463 Local error estimation 386\u003c\/p\u003e \u003cp\u003eConcluding remarks 387\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 General Linear Methods 389\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e50 RepresentingMethods in General Linear Form 389\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e500 Multivalue–multistage methods 389\u003c\/p\u003e \u003cp\u003e501 Transformations of methods 391\u003c\/p\u003e \u003cp\u003e502 Runge–Kutta methods as general linear methods 392\u003c\/p\u003e \u003cp\u003e503 Linear multistep methods as general linear methods 393\u003c\/p\u003e \u003cp\u003e504 Some known unconventional methods 396\u003c\/p\u003e \u003cp\u003e505 Some recently discovered general linear methods 398\u003c\/p\u003e \u003cp\u003e\u003cb\u003e51 Consistency, Stability and Convergence 400\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e510 Definitions of consistency and stability 400\u003c\/p\u003e \u003cp\u003e511 Covariance of methods 401\u003c\/p\u003e \u003cp\u003e512 Definition of convergence 403\u003c\/p\u003e \u003cp\u003e513 The necessity of stability 404\u003c\/p\u003e \u003cp\u003e514 The necessity of consistency 404\u003c\/p\u003e \u003cp\u003e515 Stability and consistency imply convergence 406\u003c\/p\u003e \u003cp\u003e\u003cb\u003e52 The Stability of General Linear Methods 412\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e520 Introduction 412\u003c\/p\u003e \u003cp\u003e521 Methods with maximal stability order 413\u003c\/p\u003e \u003cp\u003e522 Outline proof of the Butcher–Chipman conjecture 417\u003c\/p\u003e \u003cp\u003e523 Non-linear stability 419\u003c\/p\u003e \u003cp\u003e524 Reducible linear multistep methods and G-stability 422\u003c\/p\u003e \u003cp\u003e\u003cb\u003e53 The Order of General Linear Methods 423\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e530 Possible definitions of order 423\u003c\/p\u003e \u003cp\u003e531 Local and global truncation errors 425\u003c\/p\u003e \u003cp\u003e532 Algebraic analysis of order 426\u003c\/p\u003e \u003cp\u003e533 An example of the algebraic approach to order 428\u003c\/p\u003e \u003cp\u003e534 The underlying one-step method 429\u003c\/p\u003e \u003cp\u003e\u003cb\u003e54 Methods with Runge–Kutta stability 431\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e540 Design criteria for general linear methods 431\u003c\/p\u003e \u003cp\u003e541 The types of DIMSIM methods 432\u003c\/p\u003e \u003cp\u003e542 Runge–Kutta stability 435\u003c\/p\u003e \u003cp\u003e543 Almost Runge–Kutta methods 438\u003c\/p\u003e \u003cp\u003e544 Third order, three-stage ARK methods 441\u003c\/p\u003e \u003cp\u003e545 Fourth order, four-stage ARK methods 443\u003c\/p\u003e \u003cp\u003e546 A fifth order, five-stage method 446\u003c\/p\u003e \u003cp\u003e547 ARK methods for stiff problems 446\u003c\/p\u003e \u003cp\u003e\u003cb\u003e55 Methods with Inherent Runge–Kutta Stability 448\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e550 Doubly companion matrices 448\u003c\/p\u003e \u003cp\u003e551 Inherent Runge–Kutta stability 450\u003c\/p\u003e \u003cp\u003e552 Conditions for zero spectral radius 452\u003c\/p\u003e \u003cp\u003e553 Derivation of methods with IRK stability 454\u003c\/p\u003e \u003cp\u003e554 Methods with property F 457\u003c\/p\u003e \u003cp\u003e555 Some non-stiff methods 458\u003c\/p\u003e \u003cp\u003e556 Some stiff methods 459\u003c\/p\u003e \u003cp\u003e557 Scale and modify for stability 460\u003c\/p\u003e \u003cp\u003e558 Scale and modify for error estimation 462\u003c\/p\u003e \u003cp\u003e\u003cb\u003e56 G-symplectic methods 464\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e560 Introduction 464\u003c\/p\u003e \u003cp\u003e561 The control of parasitism 467\u003c\/p\u003e \u003cp\u003e562 Order conditions 471\u003c\/p\u003e \u003cp\u003e563 Two fourth order methods 474\u003c\/p\u003e \u003cp\u003e564 Starters and finishers for sample methods 476\u003c\/p\u003e \u003cp\u003e565 Simulations 480\u003c\/p\u003e \u003cp\u003e566 Cohesiveness 481\u003c\/p\u003e \u003cp\u003e567 The role of symmetry 487\u003c\/p\u003e \u003cp\u003e568 Efficient starting 492\u003c\/p\u003e \u003cp\u003eConcluding remarks 497\u003c\/p\u003e \u003cp\u003eReferences 499\u003c\/p\u003e \u003cp\u003eIndex 509\u003c\/p\u003e \u003cp\u003e\u003cb\u003eJ.C Butcher\u003c\/b\u003e, \u003ci\u003eEmeritus Professor, University of Auckland, New Zealand\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eA new edition of this classic work, comprehensively revised to present exciting new developments in this important subject\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eThe 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.\u003c\/p\u003e \u003cp\u003eIn 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.\u003c\/p\u003e \u003cp\u003e\u003ci\u003eKey features\u003c\/i\u003e:                                                     \u003c\/p\u003e \u003cp\u003eœ  Presents a comprehensive and detailed study of the subject\u003c\/p\u003e \u003cp\u003eœ  Covers both practical and theoretical aspects\u003c\/p\u003e \u003cp\u003eœ  Includes widely accessible topics along with sophisticated and advanced details\u003c\/p\u003e \u003cp\u003eœ  Offers a balance between traditional aspects and modern developments\u003c\/p\u003e \u003cp\u003e \u003c\/p\u003e \u003cp\u003eThis third edition of \u003ci\u003eNumerical Methods for Ordinary Differential Equations\u003c\/i\u003e 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.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47989702656229,"sku":"NP9781119121503","price":131.95,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781119121503.jpg?v=1761785168","url":"https:\/\/k12savings.com\/es\/products\/numerical-methods-for-ordinary-differential-equations-isbn-9781119121503","provider":"K12savings","version":"1.0","type":"link"}