Preface |
Differential and Difference Equations / 1: |
Differential Equation Problems / 10: |
Differential Equation Theory / 11: |
Difference Equation Problems / 100: |
Introduction to differential equations |
Difference Equation Theory / 13: |
The Kepler problem / 2: |
Numerical Differential Equation Methods |
The Euler Method / 102: |
Many-body gravitational problems |
Analysis of the Euler Method / 21: |
A problem arising from the method of lines / 22: |
Generalizations of the Euler Method |
Runge-Kutta Methods / 104: |
The simple pendulum |
Linear Multistep Methods / 24: |
A chemical kinetics problem / 25: |
Taylor Series Methods |
Hybrid Methods / 106: |
The van der Pol equation and limit cycles |
The Lotka-Volterra problem and periodic orbits / 3: |
Preliminaries |
Order Conditions / 31: |
Low Order Explicit Methods / 32: |
Existence and uniqueness of solutions / 33: |
Runge-Kutta Methods with Error Estimates |
Implicit Runge-Kutta Methods / 111: |
Linear systems of differential equations |
Stability of Implicit Runge-Kutta Methods / 35: |
Stiff differential equations / 36: |
Implementable Implicit Runge-Kutta Methods |
Order Barriers / 113: |
Laplace transforms |
Algebraic Properties of Runge-Kutta Methods / 38: |
Implementation Issues / 39: |
Introduction to difference equations / 120: |
A linear problem / 40: |
The Order of Linear Multistep Methods |
Errors and Error Growth / 122: |
The Fibonacci difference equation |
Stability Characteristics / 43: |
Three quadratic problems / 44: |
Order and Stability Barriers |
One-leg Methods and G-stability / 124: |
Iterative solutions of a polynomial equation |
The arithmetic-geometric mean / 46: |
General Linear Methods |
Representing Methods in General Linear Form / 50: |
Consistency, Stability and Convergence / 51: |
Linear difference equations / 52: |
The Stability of General Linear Methods |
The Order of General Linear Methods / 131: |
Constant coefficients |
Methods with Runge-Kutta Stabiulity / 54: |
References |
Powers of matrices |
Index |
The Z-transform / 133: |
Introduction to the Euler method / 200: |
Some numerical experiments / 201: |
Calculations with stepsize control / 202: |
Calculations with mildly stiff problems / 203: |
Calculations with the implicit Euler method / 204: |
Formulation of the Euler method / 210: |
Local truncation error / 211: |
Global truncation error / 212: |
Convergence of the Euler method / 213: |
Order of convergence / 214: |
Asymptotic error formula / 215: |
Stability characteristics / 216: |
Local truncation error estimation / 217: |
Rounding error / 218: |
Introduction / 220: |
More computations in a step / 221: |
Greater dependence on previous values / 222: |
Use of higher derivatives / 223: |
Multistep-multistage-multiderivative methods / 224: |
Implicit methods / 225: |
Local error estimates / 226: |
Historical introduction / 230: |
Second order methods / 231: |
The coefficient tableau / 232: |
Third order methods / 233: |
Introduction to order conditions / 234: |
Fourth order methods / 235: |
Higher orders / 236: |
Implicit Runge-Kutta methods / 237: |
Numerical examples / 238: |
Adams methods / 240: |
General form of linear multistep methods / 242: |
Consistency, stability and convergence / 243: |
Predictor-corrector Adams methods / 244: |
The Milne device / 245: |
Starting methods / 246: |
Introduction to Taylor series methods / 247: |
Manipulation of power series / 251: |
An example of a Taylor series solution / 252: |
Other methods using higher derivatives / 253: |
The use of f derivatives / 254: |
Further numerical examples / 255: |
Pseudo Runge-Kutta methods / 260: |
Generalized linear multistep methods / 262: |
General linear methods / 263: |
Rooted trees / 264: |
Functions on trees / 301: |
Some combinatorial questions / 302: |
The use of labelled trees / 303: |
Differentiation / 304: |
Taylor's theorem / 305: |
Elementary differentials / 310: |
The Taylor expansion of the exact solution / 311: |
Elementary weights / 312: |
The Taylor expansion of the approximate solution / 313: |
Independence of the elementary differentials / 314: |
Conditions for order / 315: |
Order conditions for scalar problems / 316: |
Independence of elementary weights / 317: |
Methods of orders less than four / 318: |
Simplifying assumptions / 321: |
Methods of order four / 322: |
New methods from old / 323: |
Methods of order five / 324: |
Methods of order six / 325: |
Methods of orders greater than six / 326: |
Richardson error estimates / 330: |
Methods with built-in estimates / 332: |
A class of error-estimating methods / 333: |
The methods of Fehlberg / 334: |
The methods of Verner / 335: |
The methods of Dormand and Prince / 336: |
Solvability of implicit equations / 340: |
Methods based on Gaussian quadrature / 342: |
Reflected methods / 343: |
Methods based on Radau and Lobatto quadrature / 344: |
A-stability, A([alpha])-stability and L-stability / 350: |
Criteria for A-stability / 351: |
Pade approximations to the exponential function / 352: |
A-stability of Gauss and related methods / 353: |
Order stars / 354: |
Order arrows and the Ehle barrier / 355: |
AN-stability / 356: |
Non-linear stability / 357: |
BN-stability of collocation methods / 358: |
The V and W transformations / 359: |
Implementation of implicit Runge-Kutta methods / 360: |
Diagonally-implicit Runge-Kutta methods / 361: |
The importance of high stage order / 362: |
Singly implicit methods / 363: |
Generalizations of singly-implicit methods / 364: |
Effective order and DESIRE methods / 365: |
Explicit barriers / 370: |
An upper bound on the required number of stages / 371: |
Motivation / 380: |
Equivalence classes of Runge-Kutta methods / 381: |
The group of Runge-Kutta methods / 382: |
The Runge-Kutta group / 383: |
A homomorphism between two groups / 384: |
A generalization of G[subscript 1] / 385: |
Recursive formula for the product / 386: |
Some special elements of G / 387: |
Some subgroups and quotient groups / 388: |
An algebraic interpretation of effective order / 389: |
Optimal sequences / 390: |
Acceptance and rejection of steps / 392: |
Error per step versus error per unit step / 393: |
Control theoretic considerations / 394: |
Solving the implicit equations / 395: |
Fundamentals / 400: |
Convergence / 401: |
Stability / 403: |
Consistency / 404: |
Necessity of conditions for convergence / 405: |
Sufficiency of conditions for convergence / 406: |
Criteria for order / 410: |
Derivation of methods / 411: |
Backward difference methods / 412: |
Further remarks on error growth / 420: |
The underlying one-step method / 422: |
Weakly stable methods / 423: |
Variable stepsize / 424: |
Stability regions / 430: |
Examples of the boundary locus method / 432: |
Examples of the Schur criterion / 433: |
Stability of predictor-corrector methods / 434: |
Survey of barrier results / 440: |
Maximum order for a convergent k step method / 441: |
Order stars for linear multistep methods / 442: |
Order arrows for linear multistep methods / 443: |
The one-leg counterpart to a linear multistep method / 450: |
The concept of G-stability / 451: |
Transformations relating one-leg and linear multistep methods / 452: |
Effective order interpretation / 453: |
Concluding remarks on G-stability / 454: |
Survey of implementation considerations / 460: |
Representation of data / 461: |
Variable stepsize for Nordsieck methods / 462: |
Local error estimation / 463: |
Multivalue multistage methods / 500: |
Transformations of methods / 501: |
Runge-Kutta methods as general linear methods / 502: |
Linear multistep methods as general linear methods / 503: |
Some known unconventional methods / 504: |
Some recently discovered general linear methods / 505: |
Definitions of consistency and stability / 510: |
Covariance of methods / 511: |
Definition of convergence / 512: |
The necessity of stability / 513: |
The necessity of consistency / 514: |
Stability and consistency imply convergence / 515: |
Methods with maximal stability order / 520: |
Possible definitions of order / 530: |
Algebraic analysis of order / 531: |
An example of the algebraic approach to order / 532: |
Methods with Runge-Kutta stability / 533: |
Design criteria for general linear methods / 540: |
The types of DIMSIM methods / 541: |
Runge-Kutta stability / 542: |
Almost Runge-Kutta methods / 543: |
Fourth order, four stage ARK methods / 544: |
Doubly companion matrices / 545: |
Inherent Runge-Kutta stability / 546: |
Derivation of methods with IRK stability / 547: |
Some nonstiff methods / 548: |
Some stiff methods / 549: |
Preface |
Differential and Difference Equations / 1: |
Differential Equation Problems / 10: |
Differential Equation Theory / 11: |
Difference Equation Problems / 100: |
Introduction to differential equations |