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H 1 High-order Magnus integrators for non-autonomous linear evolution equations

Auteurs : Thalhammer, Mechthild (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : The class of commutator-free Magnus integrators is known to provide a favourable alternative to standard interpolatory Magnus integrators, in particular for large-scale applications arising in the time integration of non-autonomous linear evolution equations. A high-order commutator-free Magnus integrator is given by a composition of several exponentials that comprise certain linear combinations of the values of the defining operator at specified nodes. Due to the fact that previously proposed commutator-free Magnus integrators of order five or higher involve negative coefficients in the linear combinations, severe instabilities are observed for spatially semi-discretised partial differential equations of parabolic type or for master equations describing dissipative quantum systems, respectively. In order to remedy this issue, two different approaches for the design of efficient Magnus integrators of orders four, five, and six are pursued: (i) the study of commutator-free Magnus integrators involving complex coefficients with positive real part, and (ii) the study of unconventional Magnus integrators that comprise in addition a single exponential involving a commutator. Numerical experiments for test equations of Schrödinger and parabolic type confirm that the identified novel Magnus integrators are superior to Magnus integrators previously proposed in the literature.

    Codes MSC :
    65M12 - Stability and convergence of numerical methods (IVP of PDE)
    35Q41 - Time-dependent Schrödinger equations, Dirac equations

      Informations sur la Vidéo

      Réalisateur : Hennenfent, Guillaume
      Langue : Anglais
      Date de publication : 15/07/2016
      Date de captation : 30/06/2016
      Sous collection : Research talks
      Format : MP4
      arXiv category : Numerical Analysis ; Analysis of PDEs
      Domaine : Numerical Analysis & Scientific Computing ; PDE
      Durée : 00:40:58
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2016-06-30_Thalhammer.mp4

    Informations sur la rencontre

    Nom de la rencontre : New challenges in mathematical modelling and numerical simulation of superfluids / Nouveaux défis dans la modélisation mathématique et la simulation numérique de systèmes superfluides
    Organisateurs de la rencontre : Danaila, Ionut ; Carles, Rémi ; Weizhu, Bao
    Dates : 27/06/2016 - 01/07/2016
    Année de la rencontre : 2016
    URL Congrès : http://conferences.cirm-math.fr/1386.html

    Citation Data

    DOI : 10.24350/CIRM.V.19009403
    Cite this video as: Thalhammer, Mechthild (2016). High-order Magnus integrators for non-autonomous linear evolution equations. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19009403
    URI : http://dx.doi.org/10.24350/CIRM.V.19009403

    Voir aussi


    1. Alvermann, A., & Fehske, H. (2011). High-order commutator-free exponential time-propagation of driven quantum systems. Journal of Computational Physics, 230(15), 5930-5956 - http://dx.doi.org/10.1016/j.jcp.2011.04.006

    2. Alvermann, A., Fehske, H., & Littlewood, P.B. (2012). Numerical time propagation of quantum systems in radiation fields. New Journal of Physics, 14, 105008 - http://dx.doi.org/10.1088/1367-2630/14/10/105008

    3. Blanes, S., & Moan, P.C. (2006). Fourth- and sixth-order commutator-free Magnus integrators for linear and nonlinear dynamical systems. Applied Numerical Mathematics, 56(12), 1519-1537 - http://dx.doi.org/10.1016/j.apnum.2005.11.004

    4. Blanes, S., Casas, F., Oteo, J.A., & Ros, J. (2009). The Magnus expansion and some of its applications. Physics Reports, 470(5-6), 151-238 - http://dx.doi.org/10.1016/j.physrep.2008.11.001

    5. Thalhammer, M. (2006). A fourth-order commutator-free exponential integrator for nonautonomous differential equations. SIAM Journal on Numerical Analysis, 44(2), 851-861 - http://dx.doi.org/10.1137/05063042