Theory of implicit extrapolation methods for ordinary differential equations

Proceedings of the Numerical Mathematics and Advanced Applications (ENUMATH 2001) conference. Springer-Verlag Italia, Milano, ISBN 88-470-0180-3 (2003), 909-918
http://dx.doi.org/10.1007/978-88-470-2089-4_82

In this paper we develop a general theory of implicit extrapolation methods for ordinary differential equations. We demonstrate how to implement such methods in practice. We first consider the common principles of constructing extrapolation methods. After that we discuss implicit extrapolation methods. Then we derive quadratic extrapolation for implicit one-step methods possessing a quadratic asymptotic expansion of the global error Finally, we present a brief outline of a theory of minimally implicit methods that extends the concept of linearly implicit methods to arbitiary iterative methods. In addition, the paper gives numerical examples which clearly confirm all theoretical results.