Numerical Methods for Ordinary Differential Equations

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.

• 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)