DOI: https://doi.org/10.26089/NumMet.v24r431

Difference schemes with weights for modelling fluid flows in the shallow water approximation

Authors

  • Petr N. Vabishchevich
  • Mikhail M. Chernyshov

Keywords:

Euler system
conservation laws
shallow water approximation
finite element method
two–level schemes

Abstract

Mathematical modelling of free boundary fluid flows is often based on the shallow water approximation. The system of equations includes a scalar advection equation for fluid height and a vector advection equation for velocity. The paper presents an approximation of the initial boundary value problem using the standard spatial finite element method. Time weights are used in implicit two-level schemes. The computational method is based on Newton’s method. The fulfillment of the laws of conservation of mass and total mechanical energy at the continuous and discrete level is discussed. Numerical results demonstrate the effectiveness of implicit schemes for approximating solutions to one- and two-dimensional model problems, specifically dam failure. It is shown that increasing the weight in the two-level scheme improves the monotonicity of the approximate solution.

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Published

2023-12-10

Issue

Vol. 24 (2023): Issue 4.

Section

Methods and algorithms of computational mathematics and their applications

Author Biographies

Petr N. Vabishchevich

Lomonosov Moscow State University,
Faculty of Computational Mathematics and Cybernetics
• Professor

Mikhail M. Chernyshov

Sarov Branch of Moscow State University,
• Student

References

  1. G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge Univ. Press, Cambridge, 2000).
  2. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics , Vol. 6: Fluid Mechanics (Nauka, Moscow, 1986; Butterworth-Heinemann, Oxford, 1987).
  3. S. K. Godunov and E. I. Romenskii, Elements of Continuum Mechanics and Conservation Laws (Nauchnaya Kniga, Novosibirsk, 1998; Springer, New York, 2003).
    doi 10.1007/978-1-4757-5117-8
  4. R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems (Cambridge Univ. Press, Cambridge, 2002).
  5. J. D. Anderson, Computational Fluid Dynamics: The Basics with Applications (McGraw-Hill, New York, 1995).
  6. P. Wesseling, Principles of Computational Fluid Dynamics (Springer, Berlin, 2001).
  7. N. D. Katopodes, Free-Surface Flow: Shallow-Water Dynamics (Butterworth-Heinemann, Oxford, 2018).
  8. C. B. Vreugdenhil, Numerical Methods for Shallow-Water Flow (Springer, Dordrecht, 1994).
  9. P.-L. Lions, Mathematical Topics in Fluid Mechanics , Vol. 2: Compressible Models (Oxford Univ. Press, New York, 1998).
  10. E. Feireisl, T. G. Karper, and M. Pokorn’y, Mathematical Theory of Compressible Viscous Fluids: Analysis and Numerics (Springer, Cham, 2016).
  11. K. W. Morton, Numerical Solution of Convection-Diffusion Problems (Chapman & Hall, London, 1996).
  12. A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Convection-Diffusion Problems (Editorial URSS, Moscow, 2004) [in Russian].
  13. A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Fizmatlit, Moscow, 2001; CRC Press, Boca Raton, 2001).
  14. W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations (Springer, Berlin, 2003).
    doi 10.1007/978-3-662-09017-6
  15. D. Kuzmin, A Guide to Numerical Methods for Transport Equations (University Erlangen—Nürnberg, Nürnberg, 2010).
    https://pdf4pro.com/view/a-guide-to-numerical-methods-for-transport-equations-566b18.html . Cited December 1, 2023.
  16. A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1989; CRC Press, Boca Raton, 2001).
    doi 10.1201/9780203908518
  17. H. K. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics: The Finite Volume Method (Prentice-Hall, Harlow, 2007).
  18. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements (Springer, New York, 2004).
    doi 10.1007/978-1-4757-4355-5
  19. M. G. Larson and F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications (Springer, Berlin, 2013).
    doi 10.1007/978-3-642-33287-6
  20. J. Donea and A. Huerta, Finite Element Methods for Flow Problems (Wiley, Chichester, 2003).
    doi 10.1002/0470013826
  21. O. C. Zienkiewicz, R. L. Taylor, and P. Nithiarasu, The Finite Element Method for Fluid Dynamics (Butterworth-Heinemann, Oxford, 2013).
    doi 10.1016/C2009-0-26328-8
  22. P. N. Vabishchevich, “Two-Level Schemes for the Advection Equation,” J. Comput. Phys. 363, 158-177 (2018).
    doi 10.1016/j.jcp.2018.02.044
  23. P. N. Vabishchevich, Numerical Methods for Solving Non-Stationary Problems (Editorial URSS, Moscow, 2021) [in Russian].
  24. U. M. Ascher, Numerical Methods for Evolutionary Differential Equations (SIAM Press, Philadelphia, 2008).
  25. R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (SIAM Press, Philadelphia, 2007).
  26. A. A. Samarskii, P. P. Matus, and P. N. Vabishchevich, Difference Schemes with Operator Factors (Springer, Dordrecht, 2002).
  27. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems (Springer, Berlin, 2006).
  28. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods (Springer, New York, 2008).
  29. P. N. Vabishchevich, “Decoupling Schemes for Predicting Compressible Fluid Flows,” Computers & Fluids 171, 94-102 (2018).
    doi 10.1016/j.compfluid.2018.06.012
  30. FEniCS Project.
    https://fenicsproject.org/.Cited December 2, 2023.
  31. E. F. Toro, Shock-Capturing Methods for Free-Surface Shallow Flows (Wiley, Hoboken, 2001).

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Copyright (c) 2023 П. Н. Вабищевич, М. М. Чернышов

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