Skip to main content
SpringerLink
Log in
Book cover

International Conference on Large-Scale Scientific Computing

LSSC 2019: Large-Scale Scientific Computing pp 557–565Cite as

Adaptive Exponential Integrators for MCTDHF

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11958))

Abstract

We compare exponential-type integrators for the numerical time-propagation of the equations of motion arising in the multi-configuration time-dependent Hartree-Fock method for the approximation of the high-dimensional multi-particle Schrödinger equation. We find that among the most widely used integrators like Runge-Kutta, exponential splitting, exponential Runge-Kutta, exponential multistep and Lawson methods, exponential Lawson multistep methods with one predictor/corrector step provide optimal stability and accuracy at the least computational cost, taking into account that the evaluation of the nonlocal potential terms is by far the computationally most expensive part of such a calculation. Moreover, the predictor step provides an estimator for the time-stepping error at no additional cost, which enables adaptive time-stepping to reliably control the accuracy of a computation.

Supported by the Vienna Science and Technology Fund (WWTF) grant MA14-002. The computations have been conducted on the Vienna Scientific Cluster (VSC).

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Note that in exponential integrators, the explicit time-dependence in the potential does not call for a special treatment in the numerical quadrature, in the splitting methods, the potential is propagated by an explicit Runge-Kutta method of appropriate order.

  2. 2.

    In this experiment, all multistep methods are started by the Krogstad exponential Runge-Kutta method with stepsize h/50.

References

  1. Calvo, M., Palencia, C.: A class of explicit multistep exponential integrators for semilinear problems. Numer. Math 102, 367–381 (2011)

    Article  MathSciNet  Google Scholar 

  2. Certaine, J.: The solution of ordinary differential equations with large time constants. In: Ralston, A., Wilf, H. (eds.) Mathematical Methods for Digital Computers, pp. 128–132. Wiley, Hoboken (1960)

    Google Scholar 

  3. Frenkel, J.: Advanced General Theory Wave Mechanics. Clarendon Press, Oxford (1934)

    MATH  Google Scholar 

  4. Friedli, A.: Verallgemeinerte Runge-Kutta Verfahren zur Loesung steifer Differentialgleichungssysteme. In: Bulirsch, R., Grigorieff, R.D. (eds.) Numerical Treatment of Differential Equations. Lecture Notes in Mathematics, vol. 631, pp. 35–50. Springer, Heidelberg (1978). https://doi.org/10.1007/BFb0067462

    Chapter  Google Scholar 

  5. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer-Verlag, Heidelberg (2002)

    Book  Google Scholar 

  6. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)

    Article  MathSciNet  Google Scholar 

  7. Hochbruck, M., Ostermann, A.: On the convergence of Lawson methods for semilinear stiff problems. CRC Preprint 2017/9, KIT Karlsruhe Institute of Technology (2017). https://www.waves.kit.edu/downloads/CRC1173_Preprint_2017-9.pdf

  8. Koch, O.: Convergence of exponential Lawson-multistep methods for the MCTDHF equations, to appear in M2AN. Model. Numer. Anal. 53, 2109–2119 (2019)

    Article  Google Scholar 

  9. Koch, O., Neuhauser, C., Thalhammer, M.: Error analysis of high-order splitting methods for nonlinear evolutionary Schrödinger equations and application to the MCTDHF equations in electron dynamics. M2AN Math. Model. Numer. Anal. 47, 1265–1284 (2013)

    Article  Google Scholar 

  10. Lawson, J.: Generalized Runge-Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4, 372–380 (1967)

    Article  MathSciNet  Google Scholar 

  11. Norsett, S.P.: An A-stable modification of the Adams-Bashforth methods. In: Morris, J.L. (ed.) Conference on the Numerical Solution of Differential Equations. LNM, vol. 109, pp. 214–219. Springer, Heidelberg (1969). https://doi.org/10.1007/BFb0060031

    Chapter  Google Scholar 

  12. Zanghellini, J., Kitzler, M., Brabec, T., Scrinzi, A.: Testing the multi-configuration time-dependent Hartree-Fock method. J. Phys. B: At. Mol. Phys. 37, 763–773 (2004)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040, Vienna, Austria

    Winfried Auzinger & Alexander Grosz

  2. Institute of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Harald Hofstätter & Othmar Koch

Authors
  1. Winfried Auzinger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Grosz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Harald Hofstätter
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Othmar Koch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Winfried Auzinger .

Editor information

Editors and Affiliations

  1. Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Lirkov

  2. Bulgarian Academy of Sciences, Sofia, Bulgaria

    Svetozar Margenov

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Auzinger, W., Grosz, A., Hofstätter, H., Koch, O. (2020). Adaptive Exponential Integrators for MCTDHF. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2019. Lecture Notes in Computer Science(), vol 11958. Springer, Cham. https://doi.org/10.1007/978-3-030-41032-2_64

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-41032-2_64

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-41031-5

  • Online ISBN: 978-3-030-41032-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation