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Convergence of Lax–Friedrichs and Godunov schemes for a nonstrictly hyperbolic system of conservation laws arising in oil recovery

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Abstract

This paper is devoted to the compactness framework and the convergence theorem for the Lax–Friedrichs and Godunov schemes applied to a \({2 \times 2}\) system of non-strictly hyperbolic nonlinear conservation laws that arises from mathematical models for oil recovery. The presence of a degeneracy in the hyperbolicity of the system requires a careful analysis of the entropy functions, whose regularity is necessary to obtain the result. For this purpose, it is necessary to combine the classical techniques referring to a singular Euler–Poisson–Darboux equation with the compensated compactness method.

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References

  1. Altenbach H., Eremeyev V.A., Kutschke A., Naumenko K.: Conservation laws and prediction methods for stress concentration fields. Acta Mech. 218(3–4), 349–355 (2011)

    Article  MATH  Google Scholar 

  2. Asakura F., Yamazaki M.: Viscous shock profiles for 2\({\times}\)2 systems of hyperbolic conservation laws with quadratic flux functions. Math. Anal. Fluid Gas Dyn. 1631, 47–59 (2009)

    MATH  Google Scholar 

  3. Chen G.Q.: Convergence of the Lax–Friedrichs scheme for the system of equations of isentropic gas dynamics. III. Acta Math. Sci. 8(3), 243–276 (1988)

    MathSciNet  Google Scholar 

  4. Chen, G.-Q., Kan, P.T.: Hyperbolic conservation laws with umbilic degeneracy. I. Arch. Ration. Mech. Anal. 130(3), 231–276 (Chinese) (1995)

  5. Chen G.-Q., Kan P.T.: Hyperbolic conservation laws with umbilic degeneracy. Arch. Ration. Mech. Anal. 160(4), 325–354 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen Z., Huan G., Ma Y.: Computational Methods for Multiphase Flows in Porous Media. SIAM, Philadelphia (2006)

    Book  MATH  Google Scholar 

  7. Chueh K.N., Conley C.C., Smoller J.A.: Positively invariant regions for systems of nonlinear diffusion equations. Indiana Univ. Math. J. 26(2), 373–392 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  8. Courant, R., Hilbert, D.: Methods of mathematical physics. vol. II. Wiley Classics Library. Wiley, New York. Partial differential equations, Reprint of the 1962 original, A Wiley-Interscience Publication (1989)

  9. dell’Isola F., Guarascio M., Hutter K.: A variational approach for the deformation of a saturated porous solid. a second-gradient theory extending Terzaghi’s effective stress principle. Arch. Appl. Mech. 70(5), 323–337 (2000)

    Article  MATH  ADS  Google Scholar 

  10. DiPerna R.J.: Compensated compactness and general systems of conservation laws. Trans. Am. Math. Soc. 292(2), 383–420 (1985)

    Article  MathSciNet  Google Scholar 

  11. Felaco E., Rubino B., Sampalmieri R.: Global existence to the Cauchy problem for hyperbolic conservation laws with an isolated umbilic point. Q. Appl. Math. 71(4), 629–659 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Harten A., Lax P.D., van Leer B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hoff D.: Invariant regions for systems of conservation laws. Trans. Am. Math. Soc. 289(2), 591–610 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Isaacson, E., Temple, B.: Examples and classification of non-strictly hyperbolic systems of conservation laws. In: Abstracts American Mathematics Society, vol. 6 (1985)

  15. Kan, P.T.: On the Cauchy problem of a \({2\times 2}\) system of non-strictly hyperbolic conservation laws. Courant Institute of Math. Sciences, N.Y. University. (Phd Thesis, Advisor: J. Glimm) (1989)

  16. Kan P.T.: Non-strictly hyperbolic conservation laws. Appl. Math. Lett. 4(4), 83–87 (1991)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  17. Lax P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math. 7(1), 159–193 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  18. Leverett M.C., Lewis W.B. et al.: Steady flow of gas-oil-water mixtures through unconsolidated sands. Trans. AIME 142(01), 107–116 (1941)

    Article  Google Scholar 

  19. Madeo A., dell’Isola F., Darve F.: A continuum model for deformable, second gradient porous media partially saturated with compressible fluids. J. Mech. Phys. Solids 61(11), 2196–2211 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  20. Murat F.: Compacité par compensation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 5(3), 489–507 (1978)

    MathSciNet  MATH  Google Scholar 

  21. Placidi L., dell’Isola F., Ianiro N., Sciarra G.: Variational formulation of pre-stressed solid–fluid mixture theory, with an application to wave phenomena. Eur. J. Mech. A/Solids 27(4), 582–606 (2008)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  22. Quiligotti S., Maugin G.A., dell’Isola F.: Wave motions in unbounded poroelastic solids infused with compressible fluids. Z. Angew. Math. Phys. ZAMP 53(6), 1110–1138 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rubino B.: Approximate solutions to the Cauchy problem for a class of \({2\times 2}\) nonstrictly hyperbolic systems of conservation laws. Boll. Un. Mat. Ital. B (7) 8(3), 583–614 (1994)

    MathSciNet  MATH  Google Scholar 

  24. Rubino B.: Compactness framework and convergence of Lax-Friedrichs and Godunov schemes for a \({2\times 2}\) nonstrictly hyperbolic system of conservation laws. Q. Appl. Math. 53(3), 401–421 (1995)

    MathSciNet  MATH  Google Scholar 

  25. Schaeffer D.G., Shearer M.: The classification of \({2\times2}\) systems of nonstrictly hyperbolic conservation laws, with application to oil recovery. Commun. Pure Appl. Math. 40(2), 141–178 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sciarra G., dell’Isola F., Coussy O.: Second gradient poromechanics. Int. J. Solids Struct. 44(20), 6607–6629 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Serre D.: La compacité par compensation pour les systèmes hyperboliques non linéaires de deux équations à à une dimension d’espace. J. Math. Pures Appl. (9) 65(4), 423–468 (1986)

    MathSciNet  MATH  Google Scholar 

  28. Serre, D.: Systems of conservation laws. 2. Cambridge University Press, Cambridge. Geometric structures, oscillations, and initial-boundary value problems, Translated from the 1996 French original by I. N. Sneddon (2000)

  29. Smoller J.: Shock Waves and Reaction–Diffusion Equations, Volume 258 of Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Science). Springer, New York (1983)

    Google Scholar 

  30. Tartar, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear analysis and mechanics: Heriot-Watt Symposium, vol. IV, volume 39 of Res. Notes in Math., pp. 136–212. Pitman, Boston, Mass., (1979)

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Authors and Affiliations

  1. Department of Information Engineering, Computer Sciences and Mathematics, University of L’Aquila, Via Vetoio (loc. Coppito), 67100, L’Aquila, Italy

    George Noel Djoufedie, Bruno Rubino & Rosella Sampalmieri

  2. Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany

    Elisabetta Felaco

Authors
  1. George Noel Djoufedie
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  2. Elisabetta Felaco
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  3. Bruno Rubino
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  4. Rosella Sampalmieri
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Correspondence to Bruno Rubino.

Additional information

Communicated by Victor Eremeyev, Peter Schiavone and Francesco dell'Isola.

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Djoufedie, G.N., Felaco, E., Rubino, B. et al. Convergence of Lax–Friedrichs and Godunov schemes for a nonstrictly hyperbolic system of conservation laws arising in oil recovery. Continuum Mech. Thermodyn. 28, 331–349 (2016). https://doi.org/10.1007/s00161-015-0432-7

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