Skip to main content

Numerical Methods for Ordinary Differential Equations, 2nd Edition

Numerical Methods for Ordinary Differential Equations, 2nd Edition

J. C. Butcher

ISBN: 978-0-470-75376-7 March 2008 482 Pages


In recent years the study of numerical methods for solving ordinary differential equations has seen many new developments. This second edition of the author's pioneering text is fully revised and updated to acknowledge many of these developments.  It includes a complete treatment of linear multistep methods whilst maintaining its unique and comprehensive emphasis on Runge-Kutta methods and general linear methods.

Although the specialist topics are taken to an advanced level, the entry point to the volume as a whole is not especially demanding.  Early chapters provide a wide-ranging introduction to differential equations and difference equations together with a survey of numerical differential equation methods, based on the fundamental Euler method with more sophisticated methods presented as generalizations of Euler.

Features of the book include

  • Introductory work on differential and difference equations.
  • A comprehensive introduction to the theory and practice of solving ordinary differential equations numerically.
  • A detailed analysis of Runge-Kutta methods and of linear multistep methods.
  • A complete study of general linear methods from both theoretical and practical points of view.
  • The latest results on practical general linear methods and their implementation.
  • A balance between informal discussion and rigorous mathematical style.
  • Examples and exercises integrated into each chapter enhancing the suitability of the book as a course text or a self-study treatise.

Written in a lucid style by one of the worlds leading authorities on numerical methods for ordinary differential equations and drawing upon his vast experience, this new edition provides an accessible and self-contained introduction, ideal for researchers and students following courses on numerical methods, engineering and other sciences.

Preface to the first edition.

Preface to the second edition.

1. Differential and Difference Equations.

10. Differential Equation Problems.

11. Differential Equating Theory.

12. Further Evolutionary Problems.

13. Difference Equation Problems.

14. Difference Equation Theory.

2. Numerical Differential Equation Methods.

20. The Euler Method.

21. Analysis of the Euler Method.

22. Generalizations of the Euler Method.

23. Runge-Kutta Methods.

24. Linear Multistep Methods.

25. Taylor Series Methods.

26. Hybrid Methods.

27. Introduction to Implementation.

3. Runge-Kutta Methods.

30. Preliminaries.

31. Order Conditions.

32. Low Order Explicit Methods.

33. Runge-Kutta Methods with Error Estimates.

34. Implicit Runge-Kutta Methods.

35. Stability of Implicit Runge-Kutta Methods.

36. Implementable Implicit Runge-Kutta Methods.

37. Symplectic Runge-Kutta Methods.

38. Algebraic Properties of Runge-Kutta Methods.

39. Implementation Issues.

4. Linear Multistep Methods.

40. Preliminaries.

41. The Order of Linear Multistep Methods.

42. Errors and Error Growth.

43. Stability Characteristics.

44. Order and Stability Barriers.

45. One-Leg Methods and G-stability.

46. Implementation Issues.

5. General Linear Methods.

50. Representing Method in General Linear Form.

51. Consistency, Stability and Convergence.

52. The Stability of General Linear Methods.

53. The Order of General Linear Methods.

54. Methods with Runge-Kutta stability.

55. Methods with Inherent Runge-Kutta Stability.