Skip to main content
SpringerLink
Log in

Hyperbolic Conservation Laws, Integral Balance Laws and Regularity of Fluxes

  • Original Paper
  • Published:
Communications on Applied Mathematics and Computation Aims and scope Submit manuscript

Abstract

Hyperbolic conservation laws arise in the context of continuum physics, and are mathematically presented in differential form and understood in the distributional (weak) sense. The formal application of the Gauss-Green theorem results in integral balance laws, in which the concept of flux plays a central role. This paper addresses the spacetime viewpoint of the flux regularity, providing a rigorous treatment of integral balance laws. The established Lipschitz regularity of fluxes (over time intervals) leads to a consistent flux approximation. Thus, fully discrete finite volume schemes of high order may be consistently justified with reference to the spacetime integral balance laws.

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

Similar content being viewed by others

References

  1. Ben-Artzi, M., Falcovitz, J.: A second-order Godunov-type scheme for compressible fluid dynamics. J. Comput. Phys. 55, 1–32 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ben-Artzi, M., Falcovitz, J.: Generalized Riemann Problems in Computational Fluid Dynamics. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  3. Ben-Artzi, M., Li, J.Q.: Hyperbolic conservation laws: Riemann invariants and the generalized Riemann problem. Numerische Mathematik 106, 369–425 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ben-Artzi, M., Li, J.Q.: Consistency of finite volume approximations to nonlinear hyperbolic balance laws. Math. Comput. 90, 141–169 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ben-Artzi, M., Li, J.Q.: Regularity of fluxes in nonlinear hyperbolic balance laws. Commun. Appl. Math. Comput. 5, 1289–1298 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ben-Artzi, M., Li, J.Q., Warnecke, G.: A direct Eulerian GRP scheme for compressible fluid flows. J. Comput. Phys. 218, 19–43 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, G.Q., Comi, G.E., Torres, M.: Cauchy fluxes and Gauss-Green formulas for divergence measure fields over general open sets. Arch. Rat. Mech. Anal. 233, 87–166 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, G.Q., Frid, H.: Divergence-measure fields and hyperbolic conservation laws. Arch. Rat. Mech. Anal. 147, 89–118 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, G.Q., Torres, M., Ziemer, W.: Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws. Commun. Pure Appl. Math. 62, 242–304 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der Mathematischen Wissenschaften, 4th edn. Springer, Heidelberg (2016)

  11. Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)

    MATH  Google Scholar 

  12. Federer, H.: Geometric Measure Theory. Springer, Heidelberg (1969)

    MATH  Google Scholar 

  13. Godlewski, E., Raviart, P.A.: Hyperbolic Systems of Conservation Laws. Ellipses (1991)

  14. Godunov, S.K.: Finite difference methods for numerical computations of discontinuous solutions of equations of fluid dynamics. Mat. Sb. 47, 271–295 (1959)

    MathSciNet  Google Scholar 

  15. Gurtin, M.E., Martins, L.C.: Cauchy’s theorem in classical physics. Arch. Rat. Mech. Anal. 60, 305–324 (1976)

  16. Lax, P., Wendroff, B.: Systems of conservation laws. Commun. Pure Appl. Math. 13, 217–237 (1960)

    Article  MATH  Google Scholar 

  17. Lei, X., Li, J.: Transversal effects of high order numerical schemes for compressible fluid flows. Appl. Math. Mech. (Engl. Edn.) 40, 343–354 (2019)

  18. Li, J., Du, Z.: A two-stage fourth order time-accurate discretization for Lax-Wendroff type flow solvers, I. Hyperbolic conservation laws. SIAM J. Sci. Comput. 38, 3045–3069 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, J., Wang, Y.: Thermodynamical effects and high resolution methods for compressible fluid flows. J. Comput. Phys. 343, 340–354 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Qian, J., Li, J., Wang, S.: The generalized Riemann problems for compressible fluid flows: towards high order. J. Comput. Phys. 259, 358–389 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sheng, W., Zhang, Q., Zheng, Y.: A direct Eulerian GRP scheme for a blood flow model in arteries. SIAM J. Sci. Comput. 43(3), A1975–A1996 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  22. Šilhavý, M.: The existence of the flux vector and the divergence theorem for general Cauchy fluxes. Arch. Rat. Mech. Anal. 90, 195–212 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  23. Šilhavý, M.: Divergence-measure fields and Cauchy’s stress theorem. Rend. Sem. Mat. Padova 113, 15–45 (2005)

    MathSciNet  MATH  Google Scholar 

  24. Spivak, M.: A Comprehensaive Introduction to Differential Geometry, vol. I. Publish or Perish, Inc., Houston, Texas (1979)

  25. Toro, E.: Riemann Solvers and Numerical Methods for Fluid Dynamics: a Practical Introduction, 3rd edn. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  26. Wu, K., Tang, H.: A direct Eulerian GRP scheme for spherically symmetric general relativistic hydrodynamics. SIAM J. Sci. Comput. 38(3), B458–B489 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

It is a pleasure to thank C. Dafermos and M. Slemrod for many useful comments. The manuscript is not submitted to other journals for simultaneous consideration. The submitted work is original and is not published elsewhere in any form or language. The authors certify that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honorarium; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript.

Funding

The second author (the late Professor Jiequan Li) was supported by the NSFC (Nos. 11771054, 12072042, 91852207), the Sino-German Research Group Project (No. GZ1465), and the National Key Project GJXM92579. The authors have no relevant financial or non-financial interests to disclose.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, 91904, Israel

    Matania Ben-Artzi

  2. Academy for Multidisciplinary Studies, Capital Normal University, Beijing, 100048, China

    Jiequan Li

  3. State Key Laboratory for Turbulence Research and Complex System, Peking University, Beijing, 100871, China

    Jiequan Li

Authors
  1. Matania Ben-Artzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiequan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matania Ben-Artzi.

Ethics declarations

Conflict of Interest

There are no potential conflicts of interest.

Research Involving Human Participants and/or Animal

Research did not involve Human Participants and/or Animals.

Informed consent

There is full consent by the two authors.

Additional information

Dedicated to Professor Gerald Warnecke on his 65-th birthday.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ben-Artzi, M., Li, J. Hyperbolic Conservation Laws, Integral Balance Laws and Regularity of Fluxes. Commun. Appl. Math. Comput. (2023). https://doi.org/10.1007/s42967-023-00298-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s42967-023-00298-2

Keywords

Mathematics Subject Classification

Search

Navigation