Skip to main content
SpringerLink
Log in

A High-Order Discontinuous Galerkin Lagrange Projection Scheme for the Barotropic Euler Equations

  • Conference paper
  • First Online:
Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems (FVCA 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 200))

Included in the following conference series:

  • 1307 Accesses

Abstract

This work considers the barotropic Euler equations and proposes a high-order conservative scheme based on a Lagrange-Projection decomposition. The high-order in space and time are achieved using Discontinuous Galerkin (DG) and Runge-Kutta (RK) strategies. The use of a Lagrange-Projection decomposition enables the use of time steps that are not constrained by the sound speed thanks to an implicit treatment of the acoustic waves (Lagrange step), while the transport waves (Projection step) are treated explicitly. We compare our DG discretization with the recent one (Renac in Numer Math 1-27, 2016, [7]) and state that it satisfies important non linear stability properties. The behaviour of our scheme is illustrated by several test cases.

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

Access this chapter

Log in via an institution

Institutional subscriptions

References

  1. Chalons, C., Girardin, M., Kokh, S.: Large time step and asymptotic preserving numerical schemes for the gas dynamics equations with source terms. SIAM J. Sci. Comput. 35(6), A2874–A2902 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chalons, C., Girardin, M., Kokh, S.: An all-regime Lagrange-Projection like scheme for the gas dynamics equations on unstructured meshes. Commun. Comput. Phys. 20(01), 188–233 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chalons, C., Kestener, P., Kokh, S., Stauffert, M.: A large time-step and well-balanced Lagrange-Projection type scheme for the shallow-water equations. Communic. Math. Sci. 15(3), 765–788 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cockburn, B., Shu, C.W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Coquel, F., Nguyen, Q., Postel, M., Tran, Q.: Entropy-satisfying relaxation method with large time-steps for Euler IBVPs. Math. Comput. 79, 1493–1533 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jin, S., Xin, Z.P.: The relaxation schemes for systems of conservation laws in arbitrary space dimension. Comm. Pure Appl. Math. 48(3), 235–276 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Renac, F.: A robust high-order Lagrange-Projection like scheme with large time steps for the isentropic Euler equations. Numer. Math., 1–27 (2016)

    Google Scholar 

  8. Suliciu, I.: On the thermodynamics of fluids with relaxation and phase transitions. Fluids with relaxation. Internat. J. Engrg. Sci 36, 921–947 (1998)

    Google Scholar 

  9. Xing, Y., Zhang, X., Shu, C.W.: Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv. Water Resour. 33(12), 1476–1493 (2010)

    Article  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to P. Kestener, S. Kokh and F. Renac for stimulating discussions, and the “Maison de la Simulation” for providing excellent working conditions to the second author.

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 45 avenue des États-Unis, 78035, Versailles, France

    Christophe Chalons & Maxime Stauffert

Authors
  1. Christophe Chalons
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maxime Stauffert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maxime Stauffert .

Editor information

Editors and Affiliations

  1. Equipe RAPSODI, Inria Lille - Nord Europe, Villeneuve-d’Ascq, France

    Clément Cancès

  2. Commissariat à l'énergie atomique et aux énergies alternatives, Centre de Saclay, Gif-sur-Yvette, France

    Pascal Omnes

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Chalons, C., Stauffert, M. (2017). A High-Order Discontinuous Galerkin Lagrange Projection Scheme for the Barotropic Euler Equations. In: Cancès, C., Omnes, P. (eds) Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems. FVCA 2017. Springer Proceedings in Mathematics & Statistics, vol 200. Springer, Cham. https://doi.org/10.1007/978-3-319-57394-6_7

Download citation

Publish with us

Policies and ethics

Search

Navigation