Proportional Extrapolation and Symmetric One-Step Methods

Russian Journal of Numerical Analysis and Mathematical Modelling, 26(1) (2011), 49-73
http://dx.doi.org/10.1515/RJNAMM.2011.004

It is well known that the theory of asymptotic expansion of the global error of one-step methods is an important but complicated fact in the realm of numerical analysis of differential equations. It can hardly be confirmed for majority of numerical schemes and problems by practical computations. On the other hand, this theory is a fundamental tool to justify the extrapolation technique, which is one of the most efficient means to solve ordinary differential equations. Therefore, in the recent paper Kulikov [Numer. Algorithms 53: 321–342, 2010] presented his local theory of extrapolation methods, which is based on the Richardson technique only and does not require any asymptotic global error expansion. Here, we concentrate on quadratic extrapolation. We explain which property of symmetric one-step methods provides two-order growth of accuracy of the underlying method after each extrapolation step and arrive at the notion of proportional extrapolation. We also learn more about adjoint and symmetric one-step methods. In addition, we prove that the modified Aitken–Neville algorithm works for any symmetric one-step method of an arbitrary order 2s.