J Math Chem (2017) 55:1521–1547
An efﬁcient six-step method for the solution of the
Dmitriy B. Berg
· T. E. Simos
Received: 26 February 2017 / Accepted: 1 March 2017 / Published online: 22 March 2017
© Springer International Publishing Switzerland 2017
Abstract In this paper we develop an efﬁcient six-step method for the solution of the
Schrödinger equation and related problems. The characteristics of the new obtained
– It is of twelfth algebraic order.
– It has three stages.
– It has vanished phase-lag.
– It has vanished its derivatives up to order two.
– All the stages of the scheme are approximations on the point x
This method is developed for the ﬁrst time in the literature. A detailed theoretical
analysis of the method is also presented. In the theoretical analysis, a comparison with
the the classical scheme of the family (i.e. scheme with constant coefﬁcients) and
with recently developed algorithm of the family with eliminated phase-lag and its ﬁrst
T. E. Simos: Highly Cited Researcher (http://isihighlycited.com/), Active Member of the European
Academy of Sciences and Arts. Active Member of the European Academy of Sciences. Corresponding
Member of European Academy of Arts, Sciences and Humanities.
Electronic supplementary material The online version of this article (doi:10.1007/s10910-017-0742-z)
contains supplementary material, which is available to authorized users.
T. E. Simos
Group of Modern Computational Methods, Ural Federal University, 19 Mira Street,
Ekaterinburg, Russian Federation 620002
Laboratory of Computational Sciences, Department of Informatics and Telecommunications,
Faculty of Economy, Management and Informatics, University of Peloponnese, 221 00 Tripolis,
10 Konitsis Street, Amphithea - Paleon Phaleron, 175 64 Athens, Greece