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Application of a Limiterless Quasi Acoustic Scheme to Solve Two-Dimensional Shallow Water Equations with an Uneven Bottom

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The article describes the application of an original quasi-acoustic scheme for numerical solution of two-dimensional shallow-water equations with an uneven bottom. The numerical scheme partitions the linear reconstruction of the solution into small-perturbation blocks. The main advantage of the quasi-acoustic scheme is that it constructs the solution without using limiters, artificial regularizers, or any tuning parameters. The scheme is verified on a number of test and prototype problems.

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References

  1. P. Glaister, “Approximate Riemann solutions of the shallow water equations,” ASCE J. Hydraulic Eng.,26, No. 3, 293–306 (1988).

    Google Scholar 

  2. R. Salmon, “Numerical solution of the two-layer shallow water equations with bottom topography,” J. Marine Research,60, 605–638 (2002).

    Article  Google Scholar 

  3. P.-W. Li and C.-M. Fan, “Generalized finite difference method for two-dimensional shallow water equations,” Eng. Anal. with Boundary Elements, 80, 58–71 (2017).

    Article  MathSciNet  Google Scholar 

  4. H. Altai and P. Dreyfuss, “Numerical solutions for 2D depth-averaged shallow water equations,” Intern. Math. Forum,13, No. 2, 79–90 (2018).

    Article  Google Scholar 

  5. M. F. Ahmad and M. S. Sulaiman, “Deficiency of finite difference methods for capturing shock waves and wave propagation over uneven bottom seabed,” Intern. J. Eng. & Technology,7, No. 3.28, 97–101 (2018).

  6. I. M. Navon, “Finite-element simulation of the shallow water equations model on a limited-area domain,” Appl. Math. Modelling,3, 337–348 (1979).

    Article  Google Scholar 

  7. M. Kawahara and T. Umetsu, “Finite element method for moving boundary problems in river flow,” Intern. J. Numer. Methods in Fluids,6, 365–386 (1986).

    Article  Google Scholar 

  8. E. Harnet, D. Y. Le Roux, V. Legat, and F. Deleersnijder, “An efficient Eulerian finite element method for the shallow water equations,” Ocean Modelling,10, 115–136 (2005).

    Article  Google Scholar 

  9. T. Zhang, F. Fang, C. C. Pain, C. Maksimovic, P. Feng, I. M. Navon, and P. D. Bates, “Application of a three-dimensional unstructured-mesh finite-element flooding model and comparison with two-dimensional approaches,” Water Resources Management,30, No. 7, 823–841 (2016).

  10. P. Azerad, J.-L. Guermond, and B. Popov, “Well-balanced second-order approximation of the shallow water equation with continuous finite elements,” SIAM J. Numer. Analysis,55, No. 6, 3203–3224 (2017).

    Article  MathSciNet  Google Scholar 

  11. J.-L. Guermond, M. Q. De Luna, B. Popov, C. E. Kees, and M. W. Farthing, “Well-balanced second-order finite element approximation of the shallow water equations with friction,” SIAM J. Scient. Computing,40, No. 6, 3873–3901 (2018).

    Article  MathSciNet  Google Scholar 

  12. A. Bermudez and M. E. Vazquez, “Upwind methods for hyperbolic conservation laws with source terms,” Computers & Fluids,23, No. 8, 1049-1071 (1994).

  13. P. Garcia-Navarro and M. E. Vazquez, “On numerical treatment of the source terms in shallow water equations,” Computers & Fluids,38, 221–234 (2009).

    Article  MathSciNet  Google Scholar 

  14. E. F. Toro, Shock-Capturing Methods for Free–Surface Shallow Flows, Wiliey, New York (2001).

  15. Q. Liang and A. G. I. Borthwick, “Adaptive quadtree simulation of shallow flows with wet–dry fronts over complex topography,” Computers & Fluids,38, 221–234 (2009).

    Article  MathSciNet  Google Scholar 

  16. H. Yuxin, Z. Ningchuan, and P. Yuguo, “Well-balanced finite volume scheme for shallow water flooding and drying over arbitrary topography,” Eng. Appl. Comput. Fluid Mechanics,7, No. 1, 40–54 (2013).

    Google Scholar 

  17. B. Turan and K.-H. Wang, “An object-oriented overland flow solver for watershed flood inundation predictions: case study of Ulus basin, Turkey,” J. Hydrol. Hydromech.,62, No. 3, 209–217 (2014).

    Article  Google Scholar 

  18. С. Berthon and С. Chalons, “A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations,” Math. Comput.,85, No. 299, 1281–1307 (2016).

    Article  MathSciNet  Google Scholar 

  19. V. Michel-Dansac, C. Berthon, S. Clain, and F. Foucher, “A well-balanced scheme for the shallow-water equations with topography or Manning friction,” J. Comput. Physics,335, 115–154 (2017).

    Article  MathSciNet  Google Scholar 

  20. A. Kurganov, “Finite-volume schemes for shallow-water equations,” Acta Numerica, 289–351 (2018).

  21. A. Harten, P. D. Lax, and B. Van Leer, “On upstream differencing and Godunov-type schemes for hyperbolic conservation laws,” SIAM Review,25, No. 1, 35–61 (1983).

  22. E. F. Tor, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, Berlin (1999).

  23. S. K. Godunov, “Difference method for numerical calculation of discontinuous solutions of hydrodynamic equations,” Mat. Sborn.,47, No. 89, 3, 271–306 (1959).

  24. M. V. Abakumov, A. M. Galanina, V. A. Isakov, N. N. Tyurina, A. P. Favorskii, and A. B. Khrulenko, “Quasi-acoustic scheme for Euler equations in hydrodynamics,” Diff. Uravn.,47, No. 7, 1092-1098 (2011).

    MATH  Google Scholar 

  25. V. A. Isakov and A. P. Favorskii, “Qusi-acoustic scheme for Euler equations of hydrodynamics in the case of two spatial dimensions,” Mat. Modelirovanie,24, No. 12, 55-59 (2012).

    Google Scholar 

  26. V. A. Isakov, “Application of a quasi-acoustic scheme to many-dimensional gas-dynamic problems,” Comput. Math. and Modeling,25, No. 3, 334–341 (2014).

    Article  MathSciNet  Google Scholar 

  27. V. A. Isakov, “Application of a quasi-acoustic scheme to solve shallow-water equations with an uneven bottom,” Comput. Math. and Modeling,29, No. 3, 319–333 (2018).

    Article  MathSciNet  Google Scholar 

  28. A. A. Samarskii, Theory of Finite-Difference Schemes [in Russian], Nauka, Moscow (1977).

  29. A. S. Petrosyan, Additional Chapters in Shallow-Water Theory [in Russian], Facsimile, IKI RAN, Moscow (2014).

  30. O. V. Bulatov, “Analytical solutions and numerical solutions of the Saint-Venant equations for some problems of the decay of a discontinuity above a cliff and a step on the bottom,” Zh. Vych. Mat. i Mat. Fiziki,54, No. 1, 150–165 (2014).

    Google Scholar 

  31. R. Bernetti, V. A. Titarev, and E. F. Toro, “Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry,” J. Comput. Physics,227, No. 6, 3212–3243 (2008).

    Article  MathSciNet  Google Scholar 

  32. R. J. LeVeque, “Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm,” J. Comput. Physics,146, No. 1, 346–365 (1998).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia

    V. A. Isakov

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Correspondence to V. A. Isakov.

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Translated from Prikladnaya Matematika i Informatika, No. 62, 2019, pp. 34–54.

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Isakov, V.A. Application of a Limiterless Quasi Acoustic Scheme to Solve Two-Dimensional Shallow Water Equations with an Uneven Bottom. Comput Math Model 31, 25–42 (2020). https://doi.org/10.1007/s10598-020-09474-y

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  • DOI: https://doi.org/10.1007/s10598-020-09474-y

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