Numerical Methods for Ordinary Differential Equations

This new book updates the exceptionally popular Numerical Analysis of Ordinary Differential Equations.

"This book is...an indispensible reference for any researcher."
—American Mathematical Society on the First Edition

Features:

• New exercises included in each chapter.
• Author is widely regarded as the world expert on Runge-Kutta methods.
• Didactic aspects of the book have been enhanced by interspersing the text with exercises.
• Updated Bibliography.

Preface.

1. Differential and Difference Equations.

10. Differential Equation Problems.

11. Differential Equation Theory.

12. Difference Equation Problems.

13. 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.

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. Order Barriers.

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 Methods 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 Stabiulity.

References.

Index. “I enjoyed reading this book. It is well written and quite approachable.” (Mathematical Reviews, 2004c)

"…covers now all the material that is modern standard in this field… belongs into any mathematical library..." (Zentralblatt Math, Vol.1040, No.09, 2004)

John Charles Butcher ONZM is a New Zealand mathematician who specialises in numerical methods for the solution of ordinary differential equations. Butcher works on multistage methods for initial value problems, such as Runge-Kutta and general linear methods.

In recent years the study of numerical methods for solving ordinary differential equations has seen many new developments. This book is a fully revised update of the author's classic 1987 text, Numerical Analysis of Ordinary Differential Equations, and includes more material on linear multistep methods, whilst maintaining its emphasis on Runge-Kutta methods. It contains introductory material on differential and difference equations, and a comprehensive review of numerical methods and their potential applications. The review starts from the Euler method applied to simple problems and builds on these ideas to introduce increasingly complex methods and problems. The author then explores Runge-Kutta, linear multistep and general linear methods in detail.

• Provides a comprehensive introduction to numerical methods for solving ordinary differential equations.
• Features introductory material on differential and difference equations.
• Includes detailed coverage of Runge-Kutta, linear multistep, and general linear methods.
• Contains exercises integrated into each chapter, enabling use as a course text or for self-study.
  * Balances informal discussion with a rigorous mathematical style.
• Written by a leading authority on numerical methods.

Researchers and students from numerical methods, engineering and other sciences will find this book provides an accessible and self-contained introduction to numerical methods for solving ordinary differential equations. It stands out amongst other books on the subject because of the author's lucid writing style, and the integrated presentation of theory, examples, and exercises.