Local theory of extrapolation methods

Abstract

In this paper we discuss the theory of one-step extrapolation methods applied both to ordinary differential equations and to index 1 semi-explicit differential-algebraic systems. The theoretical background of this numerical technique is the asymptotic global error expansion of numerical solutions obtained from general one-step methods. It was discovered independently by Henrici, Gragg and Stetter in 1962, 1964 and 1965, respectively. This expansion is also used in most global error estimation strategies as well. However, the asymptotic expansion of the global error of one-step methods is difficult to observe in practice. Therefore we give another substantiation of extrapolation technique that is based on the usual local error expansion in a Taylor series. We show that the Richardson extrapolation can be utilized successfully to explain how extrapolation methods perform. Additionally, we prove that the Aitken-Neville algorithm works for any one-step method of an arbitrary order s, under suitable smoothness.