Skip to main content
SpringerLink
Log in

Monotonicity for Genuinely Multi-step Methods: Results and Issues From a Simple Lattice Boltzmann Scheme

  • Conference paper
  • First Online:
Finite Volumes for Complex Applications X—Volume 2, Hyperbolic and Related Problems (FVCA 2023)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 433))

Included in the following conference series:

  • 163 Accesses

Abstract

This work focuses on the convergence of a two velocities lattice Boltzmann scheme towards the weak entropic solution of a non-linear scalar conservation law. The result is proved using monotonicity properties of the corresponding multi-step Finite Difference scheme and techniques germane to Finite Volume schemes. Still, only the over-relaxation regime is straightforwardly handled and we emphasize the need for a monotonicity theory for genuinely multi-step schemes for PDEs. This analysis should take the role of the initialization routines into account.

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

Access this chapter

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 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

References

  1. Higuera, F.J., Jiménez, J.: EPL 9(7), 663 (1989)

    Google Scholar 

  2. Caetano, F., Dubois, F., Graille, B.: DCDS-S (2023). https://doi.org/10.3934/dcdss.2023072

    Article  Google Scholar 

  3. Bellotti, T., Graille, B., Massot, M.: Numer. Math. 152(1), 1–40 (2022)

    Article  MathSciNet  Google Scholar 

  4. Godlewski, E., Raviart, P.-A.: Ellipses (1991)

    Google Scholar 

  5. Bellotti, T.: arXiv preprint arXiv:2302.07558 (2023)

  6. Dellacherie, S.: Acta Appl. Math. 131, 69–140 (2014)

    Article  MathSciNet  Google Scholar 

  7. Tao, T., Teng, Z.H.: Math. Comput. 64(210), 581–589 (1995)

    Google Scholar 

  8. Cheng, S.S., Lu, Y.-F.: Comput. Math. Appl. 38(7–8), 65–79 (1999)

    MathSciNet  Google Scholar 

  9. Hundsdorfer, W., Ruuth, S.J., Spiteri, R.J.: SINUM 41(2), 605–623 (2003)

    Article  Google Scholar 

  10. Tadmor, E.: Math. Comput. 43(168), 369–381 (1984)

    Article  Google Scholar 

Download references

Acknowledgements

The author thanks his PhD advisors Benjamin Graille and Marc Massot, as well as Christophe Chalons, for the discussions on this topic.

Author information

Authors and Affiliations

  1. CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120, Palaiseau, France

    Thomas Bellotti

Authors
  1. Thomas Bellotti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Bellotti .

Editor information

Editors and Affiliations

  1. IRMA, Strasbourg, France

    Emmanuel Franck

  2. Weierstrass Institute, Berlin, Germany

    Jürgen Fuhrmann

  3. IRMA, Strasbourg, France

    Victor Michel-Dansac

  4. IRMA, Strasbourg, France

    Laurent Navoret

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Bellotti, T. (2023). Monotonicity for Genuinely Multi-step Methods: Results and Issues From a Simple Lattice Boltzmann Scheme. In: Franck, E., Fuhrmann, J., Michel-Dansac, V., Navoret, L. (eds) Finite Volumes for Complex Applications X—Volume 2, Hyperbolic and Related Problems. FVCA 2023. Springer Proceedings in Mathematics & Statistics, vol 433. Springer, Cham. https://doi.org/10.1007/978-3-031-40860-1_4

Download citation

Publish with us

Policies and ethics

Search

Navigation