Skip to main content
SpringerLink
Log in

Interaction of delta shock waves for a nonsymmetric Keyfitz–Kranzer system of conservation laws

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

In this work, the mechanism for the formation of the delta shock wave is analyzed to deal with interaction of delta shock waves and contact discontinuities for a system of Keyfitz–Kranzer type by means of analysis and solutions of Riemann problems. A set of numerical experiments are provided, illustrating the theoretical findings numerically. A brief survey of the Keyfitz–Kranzer systems as a base model of fundamental nonlinear phenomena in applications is provided aiming to shed light on the intricate wave structure for other related models of conservation laws appearing in applied sciences.

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
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Abreu, E., Perez, J.: A fast, robust, and simple Lagrangian–Eulerian solver for balance laws and applications. Comput. Math. Appl. 77, 2310–2336 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abreu, E., Lambert, W., Perez, J., Santo, A.: A new finite volume approach for transport models and related applications with balancing source terms. Math. Comput. Simul. 137, 2–28 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aw, A., Rascle, M.: Resurrection of “second order” models of traffic flow. SIAM J. Appl. Math. 60(3), 916–938 (2000)

  4. Brenier, Y.: Solutions with concentration to the Riemann problem for one-dimensional Chaplygin gas dynamics. J. Math. Fluid Mech. 7, S326–S331 (2005)

    Article  MATH  Google Scholar 

  5. Bressan, A.: Global solutions of systems of conservation laws by wave-front tracking. J. Math. Anal. Appl. 170(2), 414–432 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, G.-Q., Liu, H.: Formation of \(\delta \)-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. SIAM J. Math. Anal. 34(4), 925–938 (2006)

    MathSciNet  Google Scholar 

  7. Cheng, H.: Delta shock waves for a linearly degenerate hyperbolic system of conservation laws of Keyfitz–Kranzer type. Adv. Math. Phys. 2013, 10 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Cheng, H.: Riemann problem for one-dimensional system of conservation laws of mass, momentum and energy in zero-pressure gas dynamics. Differ. Equ. Appl. 4(4), 653–664 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Danilov, V.G., Shelkovich, V.M.: Delta-shock wave type solution of hyperbolic systems of conservation laws. Q. Appl. Math. 63(3), 401–427 (2005)

    Article  MathSciNet  Google Scholar 

  10. De la cruz, R., Galvis, J., Juajibioy, J.C., Rendón, L.: Delta shock wave for a \(3 \times 3\) hyperbolic system of conservation laws. Bull. Braz. Math. Soc. New Ser. 47(1), 277–290 (2016)

  11. De la cruz, R., Santos, M.: Delta shock wave for a system of Keyfitz–Kranzer type. ZAMM Z. Angew Math. Mech. 99(3), e201700251 (2019)

  12. Ercole, G.: Delta-shock waves as self-similar viscosity limits. Q. Appl. Math. LVIII(1), 177–199 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 95–105 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  14. Godunov, S.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. 89, 271–306 (1959)

    MathSciNet  MATH  Google Scholar 

  15. Holden, H., Risebro, N.H.: Front Tracking for Hyperbolic Conservation Laws. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  16. Hwang, J., Shin, M., Shin, S., Hwang, W.: Two dimensional Riemann problem for a \(2\times 2\) system of hyperbolic conservation laws involving three constant states. Appl. Math. Comput. 321(15), 49–62 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Joseph, K.T.: A Riemann problem whose viscosity solutions contain delta-measures. Asymptot. Anal. 7, 105–120 (1993)

    Article  MATH  Google Scholar 

  18. Krejic’, N., Krunic’, T., Nedeljkov, M.: Numerical verification of delta shock waves for pressureless gas dynamics. J. Math. Anal. Appl. 345, 243–257 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Korchinski, D.: Solution of a Riemann problem for a \(2 \times 2\) system of conservation laws possessing no classical weak solution. PhD thesis, Adelphi University (1977)

  20. Floch, L., Liu, T.-P.: Generalized monotone schemes, discrete paths of extrema, and discrete entropy conditions. Math. Comput. 68(227), 1025–1055 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. LeVeque, R.J.: The dynamics of pressureless dust clouds and delta waves. J. Hyper. Differ. Equ. 1(2), 315–327 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, J.-Q., Tang, H.-Z., Warnecke, G., Zhang, L.M.: Local oscillations in finite difference solutions of hyperbolic conservation laws. Math. Comput. 78(268), 1997–2018 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu, J., Zhou, T., Yang, H.: Interaction of waves involving delta-shocks in the Chaplygin gas equations. Southeast Asian Bull. Math. 38, 825–846 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Lu, Y.-G.: Existence of global bounded weak solutions to nonsymmetric systems of Keyfitz–Kranzer type. J. Funct. Anal. 261(10), 2797–2815 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nedeljkov, M.: Delta and singular delta locus for one dimensional systems of conservation laws. Math. Methods Appl. Sci. 27(8), 931–955 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nedeljkov, M., Oberguggenberger, M.: Interactions of delta shock waves in a strictly hyperbolic system of conservation laws. J. Math. Anal. Appl. 344(2), 1143–1157 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pang, Y., Zhang, Y., Wena, Y.: Delta shock wave to the compressible fluid flow with the generalized Chaplygin gas. Int. J. Non-Linear Mech. 99, 311–318 (2018)

    Article  Google Scholar 

  28. Panov, E., Shelkovich, V.M.: \(\delta ^{\prime }\)-Shock waves as a new type of solutions to systems of conservation laws. J. Differ. Equ. 228(1), 49–86 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Perez, J.: Lagrangian–Eulerian approximation methods for balance laws and hyperbolic conservation laws. Ph.D. thesis, IMECC-Unicamp (2015)

  30. Shao, Z.Z.: The Riemann problem for the relativistic full Euler system with generalized Chaplygin proper energy density-pressure relation. Angew. Math. Phys. 69, 44 (2018). https://doi.org/10.1007/s00033-018-0937-6

    Article  MathSciNet  MATH  Google Scholar 

  31. Shelkovich, V.M.: The Riemann problem admitting \(\delta \)-, \(\delta ^{\prime }\)-shocks, and vacuum states (the vanishing viscosity approach). J. Differ. Equ. 231(2), 459–500 (2006)

    MathSciNet  MATH  Google Scholar 

  32. Shen, C.: Delta shock wave solution for a symmetric Keyfitz–Kranzer system. Appl. Math. Lett. 77, 35–43 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shen, C., Sheng, W., Sun, M.: The asymptotic limits of solutions to the Riemann problem for the scaled Leroux system. Commun. Pure Appl. Anal. 17(2), 391–411 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shen, C., Sun, M.: Interactions of delta shock waves for the transport equations with split delta functions. J. Math. Anal. Appl. 351, 747–755 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sun, M.: Interactions of elementary waves for the Aw–Rascle model. SIAM J. Appl. Math. 69, 1542–1558 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sun, M.: Interactions of delta shock waves for the chromatography equations. Applied Mathematics Letters 26, 631–637 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tadmor, E.: Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comput. 43(168), 369–381 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  38. Tan, D., Zhang, T., Zheng, Y.: Delta shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws. J. Differ. Equ. 112(1), 1–32 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  39. Yang, H., Zhang, Y.: New developments of delta shock waves and its applications in systems of conservation laws. J. Differ. Equ. 252(11), 5951–5993 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zeldovich, Y.B.: Gravitationnal instability: an approximate theory for large density perturbations. Astron. Astrophys. 5, 84–89 (1970)

    Google Scholar 

  41. Yang, H., Hu, R., Sun, Y.: The Riemann problem with three constant initial states for one-dimensional zero-pressure gas dynamics. Southeast Asian Bull. Math. 33, 179–187 (2009)

    MathSciNet  MATH  Google Scholar 

  42. Yang, H., Wang, J.: Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas. J. Math. Anal. Appl. 413(2), 800–820 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

M. Santos and R. De la cruz gratefully acknowledges the financial support of São Paulo Research Foundation (FAPESP) Grant #2016/19502-4. E. Abreu gratefully acknowledges the financial support of São Paulo Research Foundation (FAPESP) Grant #2019/20991-8, from National Council for Scientific and Technological Development - Brazil (CNPq) Grant #306385/2019-8, PETROBRAS - Brazil (Grant 2015/00398-0) as well as the MATHDATA - AUIP Network (https://www.mathdata.science/), Red Iberoamericana de Investigación en Matemáticas Aplicadas a Datos.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Universidad Pedagógica y Tecnológica de Colombia, Tunja, 150003, Colombia

    Richard De la cruz

  2. Department of Mathematics, IMECC, University of Campinas (UNICAMP), Campinas, SP, 13083-859, Brazil

    Marcelo Santos

  3. Department of Applied Mathematics, IMECC, University of Campinas (UNICAMP), Campinas, SP, 13083-859, Brazil

    Eduardo Abreu

Authors
  1. Richard De la cruz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcelo Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eduardo Abreu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Richard De la cruz.

Additional information

Communicated by Adrian Constantin.

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

De la cruz, R., Santos, M. & Abreu, E. Interaction of delta shock waves for a nonsymmetric Keyfitz–Kranzer system of conservation laws. Monatsh Math 194, 737–766 (2021). https://doi.org/10.1007/s00605-021-01524-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-021-01524-w

Keywords

Mathematics Subject Classification

Search

Navigation