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Hugoniot–Maslov Chain for Shock Waves in Buckley–Leverett Equations

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Abstract

In this paper, we apply the asymptotic method developed by V. P. Maslov [1] to obtain the approximated shock-type solutions of the generalized Riemann problem (GRP) to the Buckley–Leverett equation. We calculate the the Hugoniot–Maslov chain (an infinite ODE system) whose fulfillment is a necessary condition that must be satisfied by the coefficients of the asymptotic expansion of the shock-type solution. Numerical simulations based on the truncated Hugoniot–Maslov chain show the efficiency of this method which captures the shock wave unlike some classical finite differences schemes. Finally, we compare the results obtained in this paper with the results obtained via the same asymptotic method, but based in a previous polynomial approximation of the Buckley–Leverett flux as explained in [2]. It was observed that the application of the asymptotic method preceded by a polynomial approximation of the flux function, does not work well for long time simulation values.

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References

  1. V. P. Maslov, Propagation of Shock Waves in Isoentropic Non-Viscous Gas, in Sovremiennie Problemi Matemátiki (Itogi Naúki i Tékhniki, VINITI Moscow, 1977), Vol. 8 [in Russian].

    Google Scholar 

  2. P. Rodríguez-Bermúdez and B. Valiño-Alonso, “the Hugoniot–Maslov chains of a shock wave in conservation law with polynomial flow,” Mathematische Nachrichten 280, 907–915 (2007).

    Article  MathSciNet  Google Scholar 

  3. R. J. LeVeque, Numerical Methods for Conservation Laws, in Lectures in Mathematics. ETH Zurich (Birkhauser Basel, 1994).

    MATH  Google Scholar 

  4. V. V. Bulatov, S. Yu. Dobrokhotov, Yu. V. Vladimirov, and V. G. Danilov, “Hugoniot and Maslov chains for solitary vortex solutions to equations of shallow water, the Hill equation, and trajectories of a “typhoon eye”,” in Proceedings of the Int. Conf. the asymptotic Methods in Mechanics (St. Petersburg, 1997).

    Google Scholar 

  5. S. Yu. Dobrokhotov, “the Hugoniot–Maslov chains for solitary vortices of the shallow water equations, I. Derivation of the chains for the case of variable Coriolis forces and reduction to the Hill equation,” Russian Journal of Mathematical Physics 6 (2), 137–173 (1999).

    MathSciNet  MATH  Google Scholar 

  6. S. Yu. Dobrokhotov, “the Hugoniot–Maslov chains for solitary vortices of the shallow water equations, II. The analysis of Solutions of the truncated Chain and an approximate description of possible trajectories of mesoscale vortices (typhoons),” Russian Journal of Mathematical Physics 6 (3), 282–313 (1999).

    MathSciNet  MATH  Google Scholar 

  7. S. Yu. Dobrokhotov, E. S. Semenov, and B. Tirozzi, “the Hugoniot–Maslov chains for singular vortical solutions to quasilinear hyperbolic systems and typhoon trajectory,” Journal of Mathematical Sciences 124 (5), 5209–5249 (2004).

    Article  MathSciNet  Google Scholar 

  8. S. Yu. Dobrokhotov and B. Tirozzi, “A perturbative theory of the evolution of the center of typhoons, Zeta Functions, Topology and Quantum Physics,” Developments in Mathematics 14, 31–50 (2005).

    Article  Google Scholar 

  9. V. P. Maslov and G. A. Omelyanov, “Conditions of Hugoniot type for infinitely narrow solutions of the simple wave equation,” Sibirsk. Mat. Zh. 24 (1983) [in Russian].

    MathSciNet  Google Scholar 

  10. V. G. Danilov and G. A. Omelyanov, “Truncation of a chain Hugoniot-type conditions for shock waves and its justification for the Hopf equation,” in Preprint ESI 502 (E. Schr\(\ddot{\text{o}}\)dinger Inst. for Math. Phys., Vienna, 1997).

    Google Scholar 

  11. V. G. Danilov, G. A. Omelyanov, and E. V. Radkevich, “Hugoniot-type conditions and weak solutions to the phase-field system,” Euro Journal of Applied Mathematics 10, 55–77 (1999).

    Article  MathSciNet  Google Scholar 

  12. V. G. Danilov and G. A. Omelyanov, “Calculation of the singularity dynamics for quadratic nonlinear hyperbolic equations. Example: the Hopf equation,” in Nonlinear Theory of Generalized Functions, Chapman & Hall/CRC Research Notes in Mathematics Series (Chapman & Hall/CRC, Boca Raton, 1999), Vol. 401, pp. 63–74.

    MathSciNet  Google Scholar 

  13. A. C. Alvarez and B. Valiño-Alonso, “the Hugoniot–Maslov chain for nonlinear wave evolution with discontinuous depth,” in Memorias IV Simposio de Matematica, Fourth Italian-Latin American Conference on Applied and Industrial Mathematics (Artes Graficas, La Habana, 2001), pp. 426–434.

    MathSciNet  MATH  Google Scholar 

  14. P. Rodríguez-Bermúdez and B. Valiño-Alonso, “the asymptotic Maslov’s method for shocks of conservation laws systems with quadratic flux,” Applicable Analysis 97 (6), 888–901 (2018).

    Article  MathSciNet  Google Scholar 

  15. S. Bernard, A. Meril, P. Rodríguez-Bermúdez, and B. Valiño-Alonso, “Obtaining shock solutions via Maslov’s theory and Colombeau algebra for conservation laws with analytical coefficients,” Novi Sad Journal of Mathematics 42, 95–116 (2012).

    MathSciNet  MATH  Google Scholar 

  16. L. Schwartz, Some Applications of the Theory of Distributions, in Lectures on Modern Mathematics (Wiley, New York, 1963), Vol. I, pp. 23–58.

    Google Scholar 

  17. F V. de Sousa, Captura das Ondas de Choque no Problema Generalizado de Riemann para a Equação de Buckley–Leverett Utilizando o Método Assintótico de Maslov, in Master Thesis (Universidade Federal Fluminense, Rio de Janeiro, Brazil, 2015).

    Google Scholar 

  18. R. Ravindran and P. Prasad, “A new theory of shock dynamics. Part I: Analytic considerations,” Appl. Math. Lett. 3 (2), 77–81 (1990).

    Article  MathSciNet  Google Scholar 

  19. R. Ravindran and P. Prasad, “A new theory of shock dynamics. Part II: Numerical solution,” Appl. Math. Lett. 3 (3), 107–109 (1990).

    Article  MathSciNet  Google Scholar 

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Funding

The authors acknowledge the financial support provided by the Brazilian funding agencies CAPES, FAPERJ, CNPq, and UFF-Federal Fluminense University.

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Authors and Affiliations

  1. Department of Exact Sciences, Federal Fluminense University, Volta Redonda, Rio de Janeiro, 27255125, Brazil

    P. Rodríguez-Bermúdez, D. C. Lobão & G. B. Alvarez

  2. Federal Fluminense University, Volta Redonda, Rio de Janeiro, 27255125, Brazil

    F. V. Sousa

  3. Havana University, Havana, 10400, Cuba

    B. Valiño-Alonso

Authors
  1. P. Rodríguez-Bermúdez
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  2. F. V. Sousa
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  3. D. C. Lobão
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  4. G. B. Alvarez
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  5. B. Valiño-Alonso
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Correspondence to P. Rodríguez-Bermúdez.

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Rodríguez-Bermúdez, P., Sousa, F.V., Lobão, D.C. et al. Hugoniot–Maslov Chain for Shock Waves in Buckley–Leverett Equations. Math Notes 110, 738–753 (2021). https://doi.org/10.1134/S0001434621110110

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