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Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2022, Volume 62, Number 7, Pages 1138–1157
DOI: https://doi.org/10.31857/S0044466922070110 (Mi zvmmf11424) 		 
Partial Differential Equations

Nonlinear Schrödinger equation and the hyperbolization method

A. D. Yunakovsky

Institute of Applied Physics, Russian Academy of Sciences, 603950, Nizhny Novgorod, Russia
DOI: https://doi.org/10.31857/S0044466922070110
Abstract: “Nonstandard” equations (like a nonlinear Schrödinger one) that require very small steps in space and time in numerical computations are considered. Methods for time step increase via hyperbolization, i.e., adding the second time derivative multiplied by a small parameter, are studied. It is shown that the results can be improved by introducing an additional damping term associated with the same small parameter. The limiting values for the relation between the small parameter and the stepsizes in space and time are found.
Key words: nonlinear Schrödinger equation, hyperbolization method, amplifier, FFT, nonstationary Schrödinger equation, slip-step method, optical fiber.
Funding agency Grant number
Scientific and Educational Mathematical Center Nizhny Novgorod National Research State University 075-02-2020-1632
This work was supported by the Scientific and Educational Mathematical Center of Nizhny Novgorod State University, agreement no. 075-02-2020-1632.
Received: 06.02.2021
Revised: 12.11.2021
Accepted: 11.03.2022
English version:
Computational Mathematics and Mathematical Physics, 2022, Volume 62, Issue 7, Pages 1112–1130
DOI: https://doi.org/10.1134/S0965542522070119
Bibliographic databases:
Document Type: Article
UDC: 517.95
Language: Russian
Citation: A. D. Yunakovsky, “Nonlinear Schrödinger equation and the hyperbolization method”, Zh. Vychisl. Mat. Mat. Fiz., 62:7 (2022), 1138–1157; Comput. Math. Math. Phys., 62:7 (2022), 1112–1130
Citation in format AMSBIB
\Bibitem{Yun22}
\by A.~D.~Yunakovsky
\paper Nonlinear Schr\"odinger equation and the hyperbolization method
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2022
\vol 62
\issue 7
\pages 1138--1157
\mathnet{http://mi.mathnet.ru/zvmmf11424}
\crossref{https://doi.org/10.31857/S0044466922070110}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4466935}
\elib{https://elibrary.ru/item.asp?id=48621822}
\transl
\jour Comput. Math. Math. Phys.
\yr 2022
\vol 62
\issue 7
\pages 1112--1130
\crossref{https://doi.org/10.1134/S0965542522070119}
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