Numerical Methods for Ordinary Differential Equations : Initial Value Problems Numerical Methods for Ordinary Differential Equations is a self contained introduction to a fundamental field of numerical analysis and scientific computation. Written for undergraduate students with a mathematical background this book focuses on the analysis of numerical methods without losing sight of the practical nature of the subject. It covers the topics traditionally treated in a first course but also highlights new and emerging themes. Chapters are broken down into `lecture sized pieces motivated and illustrated by numerous theoretical and computational examples. Over 200 exercises are provided and these are starred according to their degree of difficulty. Solutions to all exercises are available to authorized instructors. The book covers key foundation topics: o Taylor series methods o Runge Kutta methods o Linear multistep methods o Convergence o Stability and a range of modern themes: o Adaptive stepsize selection o Long term dynamics o Modified equations o Geometric integration o Stochastic differential equations The prerequisite of a basic university level calculus class is assumed although appropriate background results are also summarized in appendices. A dedicated website for the book containing extra information can be found via www.springer.com. ODEs—An Introduction Euler’s Method The Taylor Series Method Linear Multistep Methods—I: Construction and Consistency Linear Multistep Methods—II: Convergence and Zero Stability Linear Multistep Methods—III: Absolute Stability Linear Multistep Methods—IV: Systems of ODEs Linear Multistep Methods—V: Solving Implicit Methods Runge–Kutta Method—I: Order Conditions Runge Kutta Methods–II Absolute Stability Adaptive Step Size Selection Long Term Dynamics Modified Equations Geometric Integration Part I—Invariants Geometric Integration Part II—Hamiltonian Dynamics Stochastic Differential Equations.