Skip to main content
Springer Nature Link
Log in

Sensitivity Parameter-Independent Characteristic-Wise Well-Balanced Finite Volume WENO Scheme for the Euler Equations Under Gravitational Fields

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

Abstract

Euler equations with a gravitational source term (PDEs) admit a hydrostatic equilibrium state where the source term exactly balances the flux gradient. The property of exact preservation of the equilibria is highly desirable when the PDEs are numerically solved. Li and Xing (J Comput Phys 316:145–163, 2016) proposed a high-order well-balanced characteristic-wise finite volume weighted essentially non-oscillatory (FV-WENO) scheme for the cases of isothermal equilibrium and polytropic equilibrium. On the contrary to what was claimed, the scheme is not well-balanced. The root of the problem is the precarious effects of a non-zero sensitivity parameter in the nonlinear weights of the WENO polynomial reconstruction procedure (WENO operator). The effects are identified in the theoretical proof for the well-balanced scheme and verified numerically on a coarse mesh resolution and a long time simulation of the PDEs. In this study, two simple yet effective numerical techniques derived from the multiplicative-invariance (MI) property of a WENO operator are invoked to rectify the sensitivity parameter’s dependency yielding a correct proof for the sensitivity parameter-independent (characteristic-wise) well-balanced FV-WENO scheme. The (non-)well-balanced nature of the schemes is demonstrated with several one- and two-dimensional benchmark steady state problems and a small perturbation over the steady state problems. Moreover, the one-dimensional Sod problem under the gravitational field is also simulated for showing the performance of the well-balanced FV-WENO scheme in capturing shock, contact discontinuity, and rarefaction wave in an essentially non-oscillatory nature. It also indicates that the numerical scheme with the third-order Runge–Kutta time-stepping scheme should take the CFL number less than 0.5 to mitigate the Gibbs oscillations at the shock without increasing the numerical dissipation artificially in the Lax–Friedrichs numerical flux.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

Code Availability

The custom codes generated during the current study are available from the corresponding author on reasonable request.

References

  1. Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227(6), 3191–3211 (2008)

    Article  MathSciNet  Google Scholar 

  2. Botta, N., Klein, R., Langenberg, S., Lützenkirchen, S.: Well-balanced finite volume methods for nearly hydrostatic flows. J. Comput. Phys. 196(2), 539–565 (2004)

    Article  MathSciNet  Google Scholar 

  3. Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230(5), 1766–1792 (2011)

    Article  MathSciNet  Google Scholar 

  4. Castro, M.J., Parés, C.: Well-balanced high-order finite volume methods for systems of balance laws. J. Sci. Comput. 82(2), 48 (2020)

    Article  MathSciNet  Google Scholar 

  5. Chandrashekar, P., Klingenberg, C.: A second order well-balanced finite volume scheme for Euler equations with gravity. SIAM J. Sci. Comput. 37(3), B382–B402 (2015)

    Article  MathSciNet  Google Scholar 

  6. Chandrashekar, P., Zenk, M.: Well-balanced nodal discontinuous Galerkin method for Euler equations with gravity. J. Sci. Comput. 71(3), 1062–1093 (2017)

    Article  MathSciNet  Google Scholar 

  7. Don, W.S., Li, D.-M., Gao, Z., Wang, B.-S.: A characteristic-wise alternative WENO-Z finite difference scheme for solving the compressible multicomponent non-reactive flows in the overestimated quasi-conservative form. J. Sci. Comput. 82(2), 27 (2020)

    Article  MathSciNet  Google Scholar 

  8. Don, W.S., Li, R., Wang, B.-S., Wang, Y.: A novel and robust scale-invariant WENO scheme for hyperbolic conservation laws. J. Comput. Phys. https://doi.org/10.13140/RG.2.2.18304.66567

  9. Ghosh, D., Constantinescu, E.M.: Well-balanced, conservative finite difference algorithm for atmospheric flows. AIAA J. 54(4), 1370–1385 (2016)

    Article  Google Scholar 

  10. Grosheintz-Laval, L., Käppeli, R.: High-order well-balanced finite volume schemes for the Euler equations with gravitation. J. Comput. Phys. 378, 324–343 (2019)

    Article  MathSciNet  Google Scholar 

  11. Jiang, G., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Käppeli, R., Mishra, S.: A well-balanced finite volume scheme for the Euler equations with gravitation. The exact preservation of hydrostatic equilibrium with arbitrary entropy stratification. Astron. Astrophys. 587, A94 (2016)

  14. Klingenberg, C., Puppo, G., Semplice, M.: Arbitrary order finite volume well-balanced schemes for the Euler equations with gravity. SIAM J. Sci. Comput. 41(2), A695–A721 (2019)

    Article  MathSciNet  Google Scholar 

  15. 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)

  16. Li, G., Xing, Y.: Well-balanced discontinuous Galerkin methods for the Euler equations under gravitational fields. J. Sci. Comput. 67(2), 493–513 (2016)

    Article  MathSciNet  Google Scholar 

  17. Li, G., Xing, Y.: High order finite volume WENO schemes for the Euler equations under gravitational fields. J. Comput. Phys. 316, 145–163 (2016)

    Article  MathSciNet  Google Scholar 

  18. Li, G., Xing, Y.: Well-balanced discontinuous Galerkin methods with hydrostatic reconstruction for the Euler equations with gravitation. J. Comput. Phys. 352, 445–462 (2018)

    Article  MathSciNet  Google Scholar 

  19. Li, G., Xing, Y.: Well-balanced finite difference weighted essentially non-oscillatory schemes for the Euler equations with static gravitational fields. Comput. Math. Appl. 75(6), 2071–2085 (2018)

    Article  MathSciNet  Google Scholar 

  20. 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(5), 2356–2381 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  22. Shi, J., Hu, C.Q., Shu, C.-W.: A technique of treating negative weights in WENO schemes. J. Comput. Phys. 175(1), 108–127 (2002)

    Article  Google Scholar 

  23. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)

    Article  MathSciNet  Google Scholar 

  24. Shu, C.-W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51(1), 82–126 (2009)

    Article  MathSciNet  Google Scholar 

  25. 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(2), 2003–2027 (2007)

    Article  MathSciNet  Google Scholar 

  26. Wang, B.-S., Li, P., Gao, Z., Don, W.S.: An improved fifth order alternative WENO-Z finite difference scheme for hyperbolic conservation laws. J. Comput. Phys. 374, 469–477 (2018)

    Article  MathSciNet  Google Scholar 

  27. Xing, Y., Shu, C.-W.: High order well-balanced WENO scheme for the gas dynamics equations under gravitational fields. J. Sci. Comput. 54(2–3), 645–662 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Zhang, R.: Finite volume WENO schemes and applications, Doctoral thesis. University of Science and Technology of China, Hefei (2010)

  30. Zhang, R., Zhang, M.P., Shu, C.-W.: On the order of accuracy and numerical performance of two classes of finite volume WENO schemes. Commun. Comput. Phys. 9(3), 807–827 (2011)

    Article  MathSciNet  Google Scholar 

  31. 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(2), 539–565 (2002)

    Article  Google Scholar 

Download references

Acknowledgements

We want to thank Prof. G. Li at Qingdao University and Prof. Y. Xing at Ohio State University for their support by sharing the original program used in this study and giving many valuable comments during the research. We are also grateful to the reviewers for their valuable suggestions. The research is supported by the National Natural Science Foundation of China (11801383,11871443), Hebei Provincial Natural Science Foundation (A2020210047). The author (Don) also likes to thank the Ocean University of China for providing the startup funding (201712011) to support this work.

Author information

Author notes

  1. All authors have contributed equally in this scholar work.

Authors and Affiliations

  1. Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang, China

    Peng Li

  2. School of Mathematical Sciences, Ocean University of China, Qingdao, China

    Bao-Shan Wang & Wai-Sun Don

Authors
  1. Peng Li
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Bao-Shan Wang
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Wai-Sun Don
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Wai-Sun Don.

Ethics declarations

Conflict of interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work. There is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, P., Wang, BS. & Don, WS. Sensitivity Parameter-Independent Characteristic-Wise Well-Balanced Finite Volume WENO Scheme for the Euler Equations Under Gravitational Fields. J Sci Comput 88, 47 (2021). https://doi.org/10.1007/s10915-021-01562-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01562-4

Keywords