Skip to main content
SpringerLink
Log in

Well-Balanced Discontinuous Galerkin Methods for the Euler Equations Under Gravitational Fields

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Euler equations under gravitational field admit hydrostatic equilibrium state where the flux produced by the pressure is exactly balanced by the gravitational source term. In this paper, we present well-balanced Runge–Kutta discontinuous Galerkin methods which can preserve the isothermal hydrostatic balance state exactly and maintain genuine high order accuracy for general solutions. To obtain the well-balanced property, we first reformulate the source term, and then approximate it in a way which mimics the discontinuous Galerkin approximation of the flux term. Extensive one- and two-dimensional simulations are performed to verify the properties of these schemes such as the exact preservation of the hydrostatic balance state, the ability to capture small perturbation of such state, and the genuine high order accuracy in smooth regions.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R., Perthame, B.: A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25, 2050–2065 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bermudez, A., Vazquez, M.E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23, 1049–1071 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Botta, N., Klein, R., Langenberg, S., Lützenkirchen, S.: Well-balanced finite volume methods for nearly hydrostatic flows. J. Comput. Phys. 196, 539–565 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chertock, A., Cui, S., Kurganovz, A., \(\ddot{O}\)zcan, S.N., Tadmor, E.: Well-balanced central-upwind schemes for the Euler equations with gravitation. SIAM J. Sci. Comput., submitted

  5. Cockburn, B., Karniadakis, G., Shu, C.-W.: The development of discontinuous Galerkin methods. In: Cockburn, B., Karniadakis, G., Shu C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering, Part I: Overview, vol. 11, pp. 3–50. Springer (2000)

  6. Cockburn, B., Shu, C.-W.: The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Desveaux, V., Zenk, M., Berthon, C., Klingenberg, C.: A well-balanced scheme for the Euler equation with a gravitational potential. Finite Vol. Complex Appl. VII-Methods Theor. Asp. Springer Proc. Math. Stat. 77, 217–226 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Greenberg, J.M., LeRoux, A.Y.: A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33, 1–16 (1996)

    Article  MathSciNet  Google Scholar 

  9. Kappeli, R., Mishra, S.: Well-balanced schemes for the Euler equations with gravitation. J. Comput. Phys. 259, 199–219 (2014)

    Article  MathSciNet  Google Scholar 

  10. LeVeque, R.J.: Balancing source terms and flux gradients on high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146, 346–365 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. LeVeque, R.J., Bale, D.S.: Wave propagation methods for conservation laws with source terms. In: Proceedings of the 7th International Conference on Hyperbolic Problems, pp. 609–618 (1998)

  12. Luo, J., Xu, K., Liu, N.: A well-balanced symplecticity-preserving gas-kinetic scheme for hydrodynamic equations under gravitational field. SIAM J. Sci. Comput. 33, 2356–2381 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Perthame, B., Simeoni, C.: A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38, 201–231 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Shu, C.-W.: TVB uniformly high-order schemes for conservation laws. Math. Comput. 49, 105–121 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  15. Shu, C.-W.: Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9, 1073–1084 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  16. Slyz, A., Prendergast, K.H.: Time independent gravitational fields in the BGK scheme for hydrodynamics. Astron. Astrophy. Suppl. Ser. 139, 199–217 (1999)

    Article  Google Scholar 

  17. Tian, C.T., Xu, K., Chan, K.L., Deng, L.C.: A three-dimensional multidimensional gas-kinetic scheme for the Navier–Stokes equations under gravitational fields. J. Comput. Phys. 226, 2003–2027 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Xing, Y.: Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium. J. Comput. Phys. 257, 536–553 (2014)

    Article  MathSciNet  Google Scholar 

  19. Xing, Y., Shu, C.-W.: High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208, 206–227 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Xing, Y., Shu, C.-W.: High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. J. Comput. Phys. 214, 567–598 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Xing, Y., Shu, C.-W.: A new approach of high order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. Commun. Comput. Phys. 1, 100–134 (2006)

    MathSciNet  MATH  Google Scholar 

  22. Xing, Y., Shu, C.-W.: High order well-balanced WENO scheme for the gas dynamics equations under gravitational fields. J. Sci. Comput. 54, 645–662 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Xing, Y., Shu, C.-W.: A survey of high order schemes for the shallow water equations. J. Math. Study 47, 221–249 (2014)

    MathSciNet  MATH  Google Scholar 

  24. 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, 1476–1493 (2010)

    Article  Google Scholar 

  25. Xu, K.: A well-balanced gas-kinetic scheme for the shallow-water equations with source terms. J. Comput. Phys. 178, 533–562 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Xu, K., Luo, J., Chen, S.: A well-balanced kinetic scheme for gas dynamic equations under gravitational field. Adv. Appl. Math. Mech. 2, 200–210 (2010)

    MathSciNet  Google Scholar 

  27. Zingale, M., Dursi, L.J., ZuHone, J., Calder, A.C., Fryxell, B., Plewa, T., Truran, J.W., Caceres, A., Olson, K., Ricker, P.M., Riley, K., Rosner, R., Siegel, A., Timmes, F.X., Vladimirova, N.: Mapping initial hydrostatic models in Godunov codes. Astrophys. J. Suppl. Ser. 143, 539–565 (2002)

    Article  Google Scholar 

Download references

Acknowledgments

The research of the first author is supported by the National Natural Science Foundation of P.R. China (No. 11201254, 11401332) and the Project for Scientific Plan of Higher Education in Shandong Providence of P.R. China (No. J12LI08). This work was partially performed at the State Key Laboratory of Science/Engineering Computing of P.R. China by virtue of the computational resources of Professor Li Yuan’s group. The first author is also thankful to Professor Li Yuan for his kind invitation. The research of the second author is sponsored by NSF grant DMS-1216454 and ORNL. The work was partially performed at ORNL, which is managed by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Qingdao University, Qingdao, 266071, Shandong, People’s Republic of China

    Gang Li

  2. Department of Mathematics, University of California Riverside, Riverside, CA, 92521, USA

    Yulong Xing

Authors
  1. Gang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yulong Xing
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yulong Xing.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, G., Xing, Y. Well-Balanced Discontinuous Galerkin Methods for the Euler Equations Under Gravitational Fields. J Sci Comput 67, 493–513 (2016). https://doi.org/10.1007/s10915-015-0093-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0093-5

Keywords

Search

Navigation