The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods

MATHEMATICAL AND COMPUTATIONAL INTRODUCTION. Mathematical Preliminaries. Ordinary Differential Equations. Difference Equations. Numerical Approximation. Graphs and Combinatorics. THE EULER METHOD AND ITS GENERALIZATIONS. The Euler Method. Generalizations of the Euler Method. Runge-Kutta Methods. Linear Multistep Methods. Taylor Series Methods. ANALYSIS OF RUNGE-KUTTA METHODS. The Order Conditions. Low Order Explicit Methods. The Attainable Order of Explicit Methods. High Order Explicit Methods. Implicit Runge-Kutta Methods. Stability Properties of Runge-Kutta Methods. Error Propagation in Runge-Kutta Methods. Runge-Kutta Methods with Error Estimates. Algebraic Properties of Runge-Kutta Methods. GENERAL LINEAR METHODS. Introduction to General Linear Methods. Examples of General Linear Methods. Stability, Consistency and Convergence. Order of Accuracy. Stability Properties of General Linear Methods. Practical Prospects for General Linear Methods. Bibliography.Butcher, J. C. is the author of 'The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods' with ISBN 9780471910466 and ISBN 0471910465.