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Regularized Equations for Numerical Simulation of Flows in the Two-Layer Shallow Water Approximation

  • T. G. Elizarova
  • A. V. Ivanov
Article

Abstract

Regularized equations describing hydrodynamic flows in the two-layer shallow water approximation are constructed. A conditionally stable finite-difference scheme based on the finitevolume method is proposed for the numerical solution of these equations. The scheme is tested using several well-known one-dimensional benchmark problems, including Riemann problems.

Keywords

two-layer shallow water equations quasi-gasdynamic approach regularized equations finite-volume method central-difference scheme one-dimensional flows transcritical flows 

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References

  1. 1.
    F. Bouchut and T. Morales de Luna, “An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment,” ESAIM: Math. Model. Numer. Anal. 42 (4), 638–698 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    R. Abgrall and S. Karni, “Two-layer shallow water system: a relaxation approach,” SIAM J. Sci. Comput. 31 (3), 1603–1627 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M. J. Castro, J. Macias, and C. Pares, “A Q-scheme for a class of systems of coupled conservation laws with source term: Application to a two-layer 1-D shallow water system,” ESAIM: Math. Model. Numer. Anal. 35 (1), 107–127 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    M. J. Castro, A. Pardo, C. Pares, and E. F. Toro, “On some fast well-balanced first order solvers for nonconservative systems,” Math. Comput. 79 (271), 1427–1472 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    M. J. Castro, J. A. Garcia-Rodriguez, J. M. Gonzalez-Vida, J. Macias, C. Pares, and M. E. Vazquez-Cendon, “Numerical simulation of two-layer shallow water flows through channels with irregular geometry,” J. Comput. Phys. 195 (1), 202–235 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. Kurganov and G. Petrova, “Central-upwind schemes for two-layer shallow water equations,” SIAM J. Sci. Comput. 31 (3), 1742–1773 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    K. T. Mandli, “A numerical method for the two layer shallow water equations with dry states,” Ocean Model. 72, 80–91 (2013).CrossRefGoogle Scholar
  8. 8.
    A. Chertock, A. Kurganov, Z. Qu, and T. Wu, “On a three-layer approximation of two-layer shallow water equations,” Math. Model. Anal. 18 (5), 675–693 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    S. C. Chen and S. H. Peng, “Two dimensional numerical model of two-layer shallow water equations for confluence simulation,” Adv. Water Resources 29, 1608–1617 (2006).CrossRefGoogle Scholar
  10. 10.
    V. V. Ostapenko, “Numerical simulation of wave flows caused by a shoreside landslide,” J. Appl. Mech. Tech. Phys. 40 (4), 647–654 (1999).CrossRefzbMATHGoogle Scholar
  11. 11.
    V. V. Ostapenko, “Method for theoretical estimation of imbalances in nonconservative difference schemes on a shock wave,” Dokl. Akad. Nauk SSSR 295 (2), 292–297 (1987).Google Scholar
  12. 12.
    L. V. Ovsyannikov, “Two-layer shallow water model,” J. Appl Mech. Tech. Phys. 20 (2), 127–135 (1979).CrossRefGoogle Scholar
  13. 13.
    V. Yu. Lyapidevskii and V. M. Teshukov, Mathematical Models of Long Wave Propagation in an Inhomogeneous Fluid (Sib. Otd. Ross. Akad. Nauk, Novosibirsk, 2000) [in Russian].Google Scholar
  14. 14.
    V. V. Ostapenko, “Complete systems of conservation laws for two-layer shallow water models,” J. Appl. Mech. Tech. Phys. 40 (5), 796–804 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    V. V. Ostapenko, “Stable shock waves in two-layer shallow water,” J. Appl. Math. Mech. 65 (1), 89–108 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    B. N. Chetverushkin, Kinetically Consistent Schemes in Gas Dynamic (Mosk. Gos. Univ., Moscow, 1999) [in Russian].zbMATHGoogle Scholar
  17. 17.
    Yu. V. Sheretov, Continuum Dynamics under Spatiotemporal Averaging (NITs Regulyarnaya i Khaoticheskaya Dinamika, Moscow, 2009).Google Scholar
  18. 18.
    T. G. Elizarova, Quasi-Gas Dynamic Equations (Nauchnyi Mir, Moscow, 2007; Springer-Verlag, Berlin, 2009).CrossRefzbMATHGoogle Scholar
  19. 19.
    O. V. Bulatov and T. G. Elizarova, “Regularized shallow water equations and an efficient method for numerical simulation of shallow water flows,” Comput. Math. Math. Phys. 51 (1), 160–173 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    O. V. Bulatov and T. G. Elizarova, “Regularized shallow water equations for numerical simulation of flows with a moving shoreline,” Comput. Math. Math. Phys. 56 (4), 661–679 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    O. V. Bulatov, “Analytical and numerical Riemann solutions of the Saint Venant equations for forward-and backward-facing step flows,” Comput. Math. Math. Phys. 54 (1), 158–171 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yu. V. Sheretov, Regularized Fluid Dynamic Equations (Tver. Gos. Univ., Tver, 2016) [in Russian].Google Scholar
  23. 23.
    A. A. Zlotnik, “On conservative spatial discretizations of the barotropic quasi-gasdynamic system of equations with a potential body force,” Comput. Math. Math. Phys. 56 (2), 303–319 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    A. A. Samarskii and Yu. P. Popov, Finite Difference Methods for Problems in Gas Dynamics (Nauka, Moscow, 1975) [in Russian].Google Scholar
  25. 25.
    P. E. Karabut and V. V. Ostapenko, “Problem of the decay of a small-amplitude discontinuity in two-layer shallow water: First approximation,” J. Appl. Mech. Tech. Phys. 52 (5), 698–688 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    T. G. Elizarova and A. V. Ivanov, “Quasi-gas dynamic algorithm for the numerical solution of two-layer shallow water equations,” Preprint No. 691, IPM RAN (Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 2016).Google Scholar
  27. 27.
    M. V. Buntina and V. V. Ostapenko, “TVD scheme for computing open channel wave flows,” Comput. Math. Math. Phys. 48 (12), 2241–2253 (2008).MathSciNetCrossRefGoogle Scholar
  28. 28.
    B. W. Levin and M. A. Nosov, Physics of Tsunamis (Yanus-K, Moscow, 2005; Springer, Berlin, 2008).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute of Applied MathematicsRussian Academy of SciencesMoscowRussia
  2. 2.Faculty of PhysicsMoscow State UniversityMoscowRussia

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