Ordinary Differential Equations: Initial Value Problems

Abstract

It is always possible to find an integral representation for initial value problems of ordinary differential equations whenever they are explicit in the n-th derivative of some variable y with respect to some other variable t. Consequently, this chapter starts with simple integrators and extends these single step methods to multi-step methods like Taylor series, linear multi-step methods, and Runge-Kutta methods. The introduction of Butcher tableaus makes the algorithmic description of these methods more transparent. Symplectic integrators are discussed as the methods of choice to solve equations of motion in Hamiltonian systems with energy conservation. The Kepler problem serves then as a benchmark to test simple as well as symplectic integrators. It becomes transparent that due to the accumulative nature of the methodological error of non-symplectic integrators energy conservation in Hamiltonian systems is severely violated.