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Numerical Modeling of Passive Scalar Transport in Shallow Water Based on the Quasi-Gasdynamic Approach

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Abstract

A new method for the numerical solution of the passive scalar transport equation in the framework of hydrodynamic equations in the shallow water approximation is described. The method is based on previously developed quasi-gasdynamic algorithms for numerical simulation of compressible gas flows. Smoothed equations are derived, and their difference approximations, including for flows with a pollutant source, are presented. The numerical algorithms are tested as applied to one- and two-dimensional flows. As an example, the algorithm is used to solve the problem of water circulation in Lake Vallunden. The constructed approach is generalized to passive scalar transport in the case of viscous incompressible flows.

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REFERENCES

  1. A. Chertock, A. Kurganov, and G. Petrova, “Finite-volume-particle methods for models of transport of pollutant in shallow water,” J. Sci. Comput. 27, 189–199 (2006).

    Article  MathSciNet  Google Scholar 

  2. A. Chertock and A. Kurganov, “On a hybrid finite-volume-particle method,” ESAIM: M2AN 38 (6), 1071–1091 (2004).

  3. V. I. Kvon, D. V. Kvon, S. D. Zonov, et al., “Numerical calculation of flows and long-range transport of contaminants in lowland river reservoirs,” J. Appl. Mech. Tech. Phys. 44 (6), 880–884 (2003).

    Article  Google Scholar 

  4. A. I. Delis and T. Katsaounis, “A generalized relaxation method for transport and diffusion of pollutant models in shallow water,” Comput. Methods Appl. Math. 4 (4), 410–430 (2004).

    Article  MathSciNet  Google Scholar 

  5. F. Benkhaldoun, I. Elmahi, and M. Seaid, “Well-balanced finite volume schemes for pollutant transport on unstructured meshes,” J. Comput. Phys. 226, 180–203 (2007).

    Article  MathSciNet  Google Scholar 

  6. E. M. Chaabelasri, I. Elmah, R. Abdellaoui, et al., “Well balanced adaptive simulation of pollutant transport by shallow water flows: Application to the Bay of Tangier,” Int. J. Hydraulic Eng. 3 (1), 10–23 (2014).

    Google Scholar 

  7. M.-O. Bristeau and B. Perthame, “Transport of pollutant in shallow water using kinetic schemes,” ESAIM: Proc. 10, 9–21 (2001).

    Article  MathSciNet  Google Scholar 

  8. E. Audusse and M.-O. Bristeau, “Transport of pollutant in shallow water: A two time steps kinetic method,” ESAIM: M2AN 37 (2), 389–416 (2003).

  9. E. D. Fernandez-Nieto and G. Narbona-Reina, “Extension of  WAF type methods to nonhomogeneous shallow water equations with pollutant,” J. Sci. Comput. 36, 193–217 (2008).

    Article  MathSciNet  Google Scholar 

  10. V. V. Churuksaeva and M. D. Mikhailov, “Numerical modeling of the fluid flow over the bottom topography,” Vestn. Tom. Gos. Univ. Mat. Mekh. 27 (1), 51–60 (2014).

    Google Scholar 

  11. E. M. Chaabelasri, I. Elmah, R. Abdellaoui, et al., “Numerical solution of advection diffusion reaction equation coupled with shallow water equation,” Int. J. Sci. Eng. Res. 8, 256–262 (2017).

    Google Scholar 

  12. B. N. Chetverushkin, Kinetic Schemes and Quasi-Gasdynamic System of Equations (MAKS, Moscow, 2004; CIMNE, Barcelona, 2008).

  13. Yu. V. Sheretov, Continuum Dynamics under Spatiotemporal Averaging (Izhevsk, Moscow) [in Russian].

  14. T. G. Elizarova, Quasi-Gas Dynamic Equations (Nauchnyi Mir, Moscow, 2007; Springer, Berlin, 2009).

  15. A. V. Marchenko and E. G. Morozov, “Asymmetric tide in Lake Vallunden (Spitsbergen),” Nonlinear Process Geophys. Discuss 20, 935–944 (2013).https://doi.org/10.5194/npg-20-935-2013

    Article  Google Scholar 

  16. E. G. Morozov, A. V. Marchenko, K. V. Filchuk, et al., “Sea Ice evolution and internal wave generation due to a tidal jet in a frozen sea,” Appl. Ocean Res. 87, 179–191 (2019). https://doi.org/10.1016/j.apor.2019.03.024

    Article  Google Scholar 

  17. 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), 160–174 (2011).

    Article  MathSciNet  Google Scholar 

  18. Yu. V. Sheretov, Regularized Hydrodynamic Equations (Tver. Gos. Univ., Tver’, 2016) [in Russian].

  19. T. G. Elizarova and A. V. Ivanov, Preprint No. 27, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2019).

  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).

    Article  MathSciNet  Google Scholar 

  21. A. A. Zlotnik, “On construction of quasi-gasdynamic systems of equations and the barotropic system with the potential body force,” Mat. Model. 24 (4), 65–79 (2012).

    MathSciNet  MATH  Google Scholar 

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ACKNOWLEDGMENTS

We are grateful to A.A. Zlotnik for constructive comments concerning the form of the regularized equations and for fruitful ideas on the formulation of initial conditions in transport problems. We also thank E.G. Morozova for setting up the problem about Lake Vallunden and his pieces of  advice regarding the interpretation of the results.

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  1. Federal Research Center Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 125047, Moscow, Russia

    T. G. Elizarova & A. V. Ivanov

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  1. T. G. Elizarova
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  2. A. V. Ivanov
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Correspondence to T. G. Elizarova or A. V. Ivanov.

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Translated by I. Ruzanova

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Elizarova, T.G., Ivanov, A.V. Numerical Modeling of Passive Scalar Transport in Shallow Water Based on the Quasi-Gasdynamic Approach. Comput. Math. and Math. Phys. 60, 1208–1227 (2020). https://doi.org/10.1134/S0965542520070064

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