Skip to main content
SpringerLink
Log in

An impact of forcing terms on the delta shock front

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

It is fortunate to answer the question that how shocks involve in conservation laws. So, here in this paper, Riemann’s problem is considered for pressureless (zero pressure flow) Euler’s equations having discontinuous forcing terms. To obtain the delta shock solution, and the impact of the forcing term on the delta shock front, we combine the generalized Rankine–Hugoniot conditions with the characteristic method for various situations. Moreover, during this process of obtaining the Riemann solution, some absorbing phenomena are also found, such as the occurrence of the vacuum state and the disappearance of the delta shock wave, etc.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. D. Dutykh, C. Acary-Robert, D. Bresch, Mathematical modeling of powder-snow avalanche flows. Stud. Appl. Math. 127(1), 38–66 (2011)

    Article  MathSciNet  Google Scholar 

  2. P. Montgomery, T. Moodie, Two-layer gravity currents with topography. Stud. Appl. Math. 102(3), 221–266 (1999)

    Article  MathSciNet  Google Scholar 

  3. J.E. Simpson, Gravity Currents: In the Environment and the Laboratory (Cambridge University Press, Cambridge, 1999)

    Google Scholar 

  4. D. Mihalas, B.W. Mihalas, Foundations of Radiation Hydrodynamics (Courier Corporation, North Chelmsford, 2013)

    MATH  Google Scholar 

  5. J.P. Suarez, G.B. Jacobs, W.S. Don, A high-order Dirac-delta regularization with optimal scaling in the spectral solution of one-dimensional singular hyperbolic conservation laws. SIAM J. Sci. Comput. 36(4), A1831–A1849 (2014)

    Article  MathSciNet  Google Scholar 

  6. F. Bouchut, On zero pressure gas dynamics, in Advances in Kinetic Theory and Computing, vol. 22, ed. by B. Perthame (World Scientific, Singapore, 1994), pp. 171–190

    Google Scholar 

  7. L. Boudin, A solution with bounded expansion rate to the model of viscous pressureless gases. SIAM J. Math. Anal. 32(1), 172–193 (2000)

    Article  MathSciNet  Google Scholar 

  8. J. Li, T. Zhang, S. Yang, The Two-Dimensional Riemann Problem in Gas Dynamics, vol. 98 (CRC Press, Boca Raton, 1998)

    MATH  Google Scholar 

  9. W. Sheng, T. Zhang, The Riemann Problem for the Transportation Qquations in Gas Dynamics, vol. 654 (American Mathematical Society, Providence, 1999)

    Google Scholar 

  10. F. Huang, Z. Wang, Well posedness for pressureless flow. Commun. Math. Phys. 222(1), 117–146 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  11. Z. Wang, F. Huang, X. Ding, On the Cauchy problem of transportation equations. Acta Math. Appl. Sin. (Engl. Ser.) 13(2), 113–122 (1997)

    Article  MathSciNet  Google Scholar 

  12. C. Shen, M. Sun, Interactions of delta shock waves for the transport equations with split delta functions. J. Math. Anal. Appl. 351(2), 747–755 (2009)

    Article  MathSciNet  Google Scholar 

  13. T. Chang, L. Hsiao, The Riemann problem and interaction of waves in gas dynamics. NASA STI/Recon Technical Report A 90

  14. J. Smoller, Shock Waves and Reaction–Diffusion Equations, vol. 258 (Springer, Berlin, 2012)

    MATH  Google Scholar 

  15. Y. Chang, S.W. Chou, J.M. Hong, Y.C. Lin, Existence and uniqueness of Lax-type solutions to the Riemann problem of scalar balance law with singular source term. Taiwan. J. Math. 17(2), 431–464 (2013)

    Article  MathSciNet  Google Scholar 

  16. B. Fang, P. Tang, Y.G. Wang, The Riemann problem of the Burgers equation with a discontinuous source term. J. Math. Anal. Appl. 395(1), 307–335 (2012)

    Article  MathSciNet  Google Scholar 

  17. M. Sun, Formation of delta standing wave for a scalar conservation law with a linear flux function involving discontinuous coefficients. J. Nonlinear Math. Phys. 20(2), 229–244 (2013)

    Article  MathSciNet  Google Scholar 

  18. P. Tang, Y.G. Wang, Shock waves for the hyperbolic balance laws with discontinuous sources. Math. Methods Appl. Sci. 37(14), 2029–2064 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  19. S.N. Kružkov, First order quasilinear equations in several independent variables. Math. USSR-Sbornik 10(2), 217 (1970)

    Article  Google Scholar 

  20. C. Sinestrari, The Riemann problem for an inhomogeneous conservation law without convexity. SIAM J. Math. Anal. 28(1), 109–135 (1997)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. B. Temple, J. Hong, A bound on the total variation of the conserved quantities for solutions of a general resonant nonlinear balance law. SIAM J. Appl. Math. 64(3), 819–857 (2004)

    Article  MathSciNet  Google Scholar 

  23. J. Greenberg, A. Leroux, R. Baraille, A. Noussair, Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal. 34(5), 1980–2007 (1997)

    Article  MathSciNet  Google Scholar 

  24. K. Karlsen, S. Mishra, N. Risebro, Well-balanced schemes for conservation laws with source terms based on a local discontinuous flux formulation. Math. Comput. 78(265), 55–78 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  25. E. Weinan, K. Khanin, A. Mazel, Y. Sinai, Invariant measure for Burgers equation with stochastic forcing. Ann. Math. Second Ser. 151(3), 877–960 (2000)

    Article  MathSciNet  Google Scholar 

  26. A. Saichev, W. Woyczynski, Evolution of Burgers’ turbulence in the presence of external forces. J. Fluid Mech. 331, 313–343 (1997)

    Article  ADS  Google Scholar 

  27. T. Zhang, C. Shen, The shock wave solution to the Riemann problem for the Burgers equation with the linear forcing term. Appl. Anal. 95(2), 283–302 (2016)

    Article  MathSciNet  Google Scholar 

  28. P. Montgomery, T. Moodie, Jump conditions for hyperbolic systems of forced conservation laws with an application to gravity currents. Stud. Appl. Math. 106(3), 367–392 (2001)

    Article  MathSciNet  Google Scholar 

  29. G. Srinivasan, V. Sharma, A note on the jump conditions for systems of conservation laws. Stud. Appl. Math. 110(4), 391–396 (2003)

    Article  MathSciNet  Google Scholar 

  30. C. Shen, The Riemann problem for the pressureless Euler system with the Coulomb-like friction term. IMA J. Appl. Math. 81(1), 76–99 (2015)

    MathSciNet  MATH  Google Scholar 

  31. Q. Zhang, Singular solutions to the Riemann problem for the pressureless Euler equations with discontinuous source term. arXiv preprint arXiv:1712.04160 (2017)

Download references

Author information

Authors and Affiliations

  1. Indian Institute of Technology, Roorkee, India

    Ashish Tiwari & Rajan Arora

Authors
  1. Ashish Tiwari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rajan Arora
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rajan Arora.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tiwari, A., Arora, R. An impact of forcing terms on the delta shock front. Eur. Phys. J. Plus 135, 560 (2020). https://doi.org/10.1140/epjp/s13360-020-00559-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-020-00559-6

Search

Navigation