Skip to main content
SpringerLink
Log in

Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

A convergence theorem of the fractional step Lax-Friedrichs scheme and Godunov scheme for an inhomogeneous system of isentropic gas dynamics (1<γ≦5/3) is established by using the framework of compensated compactness. Meanwhile, a corresponding existence theorem of global solutions with large data containing the vacuum is obtained.

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. Zhang Tong, Guo Yu-Fa: A class of initial-value problem for systems of aerodynamics equations. Acta Math. Sinica15, 386–396 (1965)

    Google Scholar 

  2. Nishida, T.: Global solution for an initial-boundary-value problem of a quasilinear hyperbolic system. Proc. Jpn. Acad.44, 642–646 (1968)

    Google Scholar 

  3. Bahkvarov, N.: On the existence of regular solutions in the large for quasilinear hyperbolic systems. Ah. Vychisl. Mat. Fiz.10, 969–980 (1970)

    Google Scholar 

  4. Nishida, T., Smoller, J.: Solutions in the large for some nonlinear hyperbolic conservations. Commun. Pure Appl. Math.26, 183–200 (1973)

    Google Scholar 

  5. Ding Shia-shi, Chang-Tung, Wang Ching-Hua, Hsiao-Ling, Li Tsai-Chang: A study of the global solutions for quasilinear hyperbolic systems of conservation laws. Scientica Sinica16, 317–335 (1973)

    Google Scholar 

  6. Lin Longwei: A study of the global solutions for system of gas dynamics. Acta Scien. Nat. Jilin Univ.1, 96–106 (1978)

    Google Scholar 

  7. DiPerna, R. J.: Convergence of the viscosity method for isentropic gas dynamics. Commun. Math. Phys.91, 1–30 (1983)

    Google Scholar 

  8. Ding Xiaxi, Chen Gui-Qiang, Luo Peizhu: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (I), (II). Acta Math. Sci.5, 483–500, 501–540 (1985)

    Google Scholar 

  9. Chen Gui-Qiang: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (III), ibid.6, 75–120 (1986)

    Google Scholar 

  10. Ding Xiaxi, Chen Gui-Qiang, Luo Peizhu: A supplement to the papers “Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (II)-(III)”, ibid.7, (1987)

  11. Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math.18, 697–715 (1965)

    Google Scholar 

  12. Tartar, L.: Une nouvelle methode de resolution d'equations aux derivees partielles nonlineaire. Lecture Notes in Mathematics, vol.665, pp. 228–241. Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

  13. Tartar, L.: Compensated compactness and applications to partial differential equations. In: Research notes in mathematics, nonlinear analysis and mechanics. Heriot-Watt Symposium, Vol.4, (ed.) Knops, R. J. London: Pitman Press 1979

    Google Scholar 

  14. Tartar, L.: The compensated compactness method applied to systems of conservation laws. In: Systems of nonlinear partial differential equations, NATO ASI Series, Ball, J. M. (ed.). Darhecht: D. Reidel 1983

    Google Scholar 

  15. Murat, F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat.5, 489–507 (1978)

    Google Scholar 

  16. Murat, F.: Compacité par compsensation: condition nécessaire et suffisante de continuité faible sous une hypothese de rang constant. Ann. Scuola Norm. Sup. Pisa8, 69–102 (1981)

    Google Scholar 

  17. Murat, F.: L'injection du cone positif deH −1 dansW −1,q est compact pour toutq>0. J. Math. Pures Appl.6, 309–322 (1981)

    Google Scholar 

  18. Dacorogna, B.: Weak continuity and weak lower semicontinuity of nonlinear functionals. Lecture Notes in Mathematics, Vol.922, pp. 1–20. Berlin, Heidelberg, New York: 1982

  19. Lax, P.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math.7, 159–193 (1954)

    Google Scholar 

  20. Godunov, S. K.: A difference method for numerical calculation of discontinuous solutions of the equation of hydrodynamics. Mat. Sb.47, 89, 271–306 (1959)

    Google Scholar 

  21. DiPerna, R. J.: Convergence of approximate solutions to conservation laws. Arch. Rat. Mech. Anal.82, 27–70 (1983)

    Google Scholar 

  22. DiPerna, R. J.: Compensated compactness and general systems of conservation laws, preprint

  23. Ball, J. M.: Convexity conditions and existence theorem in nonlinear elasticity. Arch. Rat. Mech. Anal.63, 337–403 (1977)

    Google Scholar 

  24. Oleinik, O.: Discontinuous solutions of nonlinear differential equations. Usp. Mat. Nauk. (N.S.)12, 3–73 (1957)

    Google Scholar 

  25. Conway, E., Smoller, J.: Global solutions of the Cauchy problem for quasilinear first-order equations in several space variables. Commun. Pure Appl. Math.19, 95–105 (1966)

    Google Scholar 

  26. Kruzkov, S.: First-order quasilinear equations with several space variables. Mat. Sb.123, 228–255 (1970)

    Google Scholar 

  27. Morawetz, C. S.: On a weak solution for a transonic flow problem. Commun. Pure Appl. Math.38, 797–818 (1985)

    Google Scholar 

  28. Serre, D.: La compacite' par compensation pour les systems hyperboliques nonlineaires de deux equations a une dimension d'espace, preprint

  29. Rascle, M.: Une resultat de “Compacité par compensation á coefficients variables”, Application à l'elasticité nonlinear. C.R.A.S.302, 311–314 (1986)

    Google Scholar 

  30. Roytburd, V., Slemrod, M.: An application of the method of compensated compactness to a problem in phase transitions. Proc. of the Symposium (to appear). Year on Material Instabilities and Continuum Mechanics. Oxford: Oxford University Press

  31. Dafermos, C. M.: Solutions inL ℞ for a conservation law with memory LCDS/CCs # 87-5

  32. Liu Tai-ping: Quasilinear hyperbolic systems. Commun. Math. Phys.68, 141–172 (1979)

    Google Scholar 

  33. Ying Lung-an, Wang Ching-hua: Global solutions of the Cauchy problem for a nonhomogeneous quasilinear hyperbolic system. Commun. Pure Appl. Math.33, 579–597 (1980)

    Google Scholar 

  34. Ying Lung-an, Wang Ching-hua: Solution in the large for nonhomogeneous quasilinear hyperbolic systems of equations. J. Math. Anal.78, 440–454 (1980)

    Google Scholar 

  35. Dafermos, C. M., Hsiao, L.: Hyperbolic systems of balance law with inhomogeneity and dissipation. Indiana U. Math. J. (c)31, 471–491 (1982)

    Google Scholar 

  36. Chueh, K., Conley, C., Smoller, J.: Positively invariant regions for systems of nonlinear diffusion equations. Indiana U. Math. J.26, 373–392 (1977)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, 251 Mercer Street, 10012, New York, NY, USA

    Chen Gui-Qiang

  2. Wuhan Institute of Mathematical Sciences, Academia Sinica, 430071, Wuhan, China

    Ding Xiaxi

  3. Institute of Systems Science, Academia Sinica, 100080, Beijing, China

    Chen Gui-Qiang & Luo Peizhu

Authors
  1. Ding Xiaxi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chen Gui-Qiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luo Peizhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Jaffe

Partially supported by U.S. NSF Grant # DMS-850403

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xiaxi, D., Gui-Qiang, C. & Peizhu, L. Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics. Commun.Math. Phys. 121, 63–84 (1989). https://doi.org/10.1007/BF01218624

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01218624

Keywords

Search

Navigation