A new implicit six-step P-stable method for the numerical solution of Schrödinger equation

ABSTRACT

In this paper, we will present a new six-step P-stable method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical integration of the one-dimensional Schrödinger equation. We perform an analysis of the local truncation error of the methods for the general case and for the special case of the Schrödinger equation, where we show the decrease of the maximum power of the energy in relation to the corresponding classical methods. We also perform a periodicity analysis. In addition, we determine their periodicity regions. Finally, we compare the new methods to the corresponding classical ones and other known methods from the literature, where we show the high efficiency of the new methods.

KEYWORDS:

• Phase fitting
• Schrödinger equation
• phase-lag
• ordinary differential equations
• P-stable
• symmetric multistep methods

2000 MSC Classifications:

• 65L05
• 65L06
• 65L20

Acknowledgments

The authors wish to thank Professor Qin Sheng for his careful reading of this paper in several stages and providing valuable feedback in order to correct it. The authors also wish to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Ali Shokri http://orcid.org/0000-0003-2699-1490

Jesús Vigo-Aguiar http://orcid.org/0000-0002-1921-6579