Skip to main page content
U.S. flag

An official website of the United States government

Dot gov

The .gov means it’s official.
Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

Https

The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

Access keys NCBI Homepage MyNCBI Homepage Main Content Main Navigation
. 2021;7(1):6.
doi: 10.1007/s40819-020-00937-9. Epub 2020 Dec 19.

Adaptive Time Propagation for Time-dependent Schrödinger equations

Winfried Auzinger  1 , Harald Hofstätter  2 , Othmar Koch  2 , Michael Quell  3
Affiliations

Adaptive Time Propagation for Time-dependent Schrödinger equations

Winfried Auzinger et al. Int J Appl Comput Math. 2021.

Abstract

We compare adaptive time integrators for the numerical solution of linear Schrödinger equations where the Hamiltonian explicitly depends on time. The approximation methods considered are splitting methods, where the time variable is split off and advanced separately, and commutator-free Magnus-type methods. The time-steps are chosen adaptively based on asymptotically correct estimators of the local error in both cases. It is found that splitting methods are more efficient when the Hamiltonian naturally suggests a separation into kinetic and potential part, whereas Magnus-type integrators excel when the structure of the problem only allows to advance the time variable separately.

Keywords: Adaptive stepsize selection; Magnus-type integrators; Splitting methods; Time-dependent Schrödinger equations.

PubMed Disclaimer

References

    1. Alverman A, Fehske H. High-order commutator-free exponential time-propagation of driven quantum systems. J. Comput. Phys. 2011;230:5930–5956. doi: 10.1016/j.jcp.2011.04.006. - DOI
    1. Auzinger W, Březinová I, Hofstätter H, Koch O, Quell M. Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part II: comparisons of local error estimation and step-selection strategies for nonlinear Schrödinger and wave equations. Comput. Phys. Commun. 2019;234:55–71. doi: 10.1016/j.cpc.2018.08.003. - DOI
    1. Auzinger, W., Dubois, J., Held, K., Hofstätter, H., Jawecki, T., Kauch, A., Koch, O., Kropielnicka, K., Singh, P., Watzenböck, C.: Efficient Magnus-type integrators for Hubbard models of solar cells. Submitted
    1. Auzinger W, Herfort W. Local error structures and order conditions in terms of Lie elements for exponential splitting schemes. Opusc. Math. 2014;34:243–255. doi: 10.7494/OpMath.2014.34.2.243. - DOI
    1. Auzinger W, Hofstätter H, Ketcheson D, Koch O. Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part I: construction of optimized schemes and pairs of schemes. BIT. 2017;57:55–74. doi: 10.1007/s10543-016-0626-9. - DOI - PMC - PubMed

LinkOut - more resources