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Computational Mathematics and Mathematical Physics

, Volume 57, Issue 2, pp 318–339

# Simulation of shallow water flows with shoaling areas and bottom discontinuities

• A. I. Aleksyuk
• V. V. Belikov
Article

## Abstract

A numerical method based on a second-order accurate Godunov-type scheme is described for solving the shallow water equations on unstructured triangular-quadrilateral meshes. The bottom surface is represented by a piecewise linear approximation with discontinuities, and a new approximate Riemann solver is used to treat the bottom jump. Flows with a dry sloping bottom are computed using a simplified method that admits negative depths and preserves the liquid mass and the equilibrium state. The accuracy and performance of the approach proposed for shallow water flow simulation are illustrated by computing one- and two-dimensional problems.

## Keywords

shallow water equations finite-volume method Riemann problem discontinuous bottom surface dry-bottom areas

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## References

1. 1.
J. J. Stoker, Water Waves (Wiley, New York, 1957; Inostrannaya Literatura, Moscow, 1959).
2. 2.
N. E. Vol’tsinger and R. V. Pyaskovskii, Shallow-Water Theory: Oceanological Problems and Numerical Methods (Gidrometeoizdat, Leningrad, 1977).Google Scholar
3. 3.
A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Fizmatlit, Moscow, 2001; Chapman and Hall/CRC, London, 2001).
4. 4.
F. R. Garcia and R. Kahawita, “Numerical solution of the shallow water equations with a MacCormack type finite difference scheme,” Mathematical Modeling in Science and Technology: The 4th International Conference (Pergamon, New York, 1983), pp. 669–673.Google Scholar
5. 5.
A. I. Delis, C. P. Skeels, and S. C. Ryrie, “Evaluation of some approximate Riemann solvers for transient open channel flows,” J. Hydraulic Res. 38 (3), 217–231 (2000).
6. 6.
V. V. Belikov and A. Yu. Semenov, “Godunov method with Kolgan’s modification for the numerical solution of the 2D shallow water equations,” Proceedings of the 10th Conference of Young Researchers from the Moscow Institute of Physics and Technology (March 23–April 7, 1985); Available from VINITI, Part 1, No. 5983-85, pp. 179–214.Google Scholar
7. 7.
V. V. Belikov and A. Yu. Semenov, “Godunov method with Kolgan’s modification as applied to computing flow plans in the tailwater area of water pipes,” in Hydraulics of Road Discharge Facilities (Saratov. Politekh. Inst., Saratov, 1985), pp. 54–57 [in Russian].Google Scholar
8. 8.
V. V. Belikov, “Iteration-free Riemann solver for shallow water equations,” in Approaches to Better Performance and Reduction in Periods of Design and Building Transportation Facilities (Vsesoyuz. Nauchno-Issled. Inst. Transp. Stroit., Moscow, 1986), pp. 81–85 [in Russian].Google Scholar
9. 9.
V. V. Belikov, Candidate’s Dissertation in Mathematics and Physics (Moscow, 1987).Google Scholar
10. 10.
V. V. Belikov and A. Yu. Semenov, Preprint No. 42 (General Physics Inst._USSR Acad. Sci., Moscow, 1988).Google Scholar
11. 11.
V. V. Belikov and A. Yu. Semenov, “A Godunov-type method based on an exact solution to the Riemann problem for the shallow-water equations,” Comput. Math. Math. Phys. 37 (8), 974–986 (1997).
12. 12.
V. V. Belikov and A. Yu. Semenov, “Design of Riemann solvers for the shallow water equations,” in Computational Fluid Dynamics of Natural Flows (Fizmatlit, Moscow, 1997) [in Russian].Google Scholar
13. 13.
V. V. Belikov and A. Yu. Semenov, “A Godunov’s type method based on an exact solution to the Riemann problem for the shallow-water equations,” Proceedings of the 4th European Computational Fluid Dynamics Conference (ECCOMAS 98) (Wiley, New York, 1998), Vol. 1, Part 1, pp. 310–315.Google Scholar
14. 14.
L. Papa, “Application of the Courant–Isaacson–Rees method to solve the shallow-water hydrodynamic equations,” Appl. Math. Comput. 15 (1), 85–92 (1984).
15. 15.
P. Glaister, “A weak formulation of Roe’s approximate Riemann solver applied to the St. Venant equations,” J. Comput. Phys. 116 (1), 189–191 (1995).
16. 16.
B. Van Leer, “Towards the ultimate conservative difference scheme V: A second-order sequel to Godunov’s method,” J. Comput. Phys. 32 (1), 101–136 (1979).
17. 17.
A. Harten, B. Engquist, S. Osher, and S. B. Chakravarthy, “Uniformly high order accurate nonoscillatory schemes III,” J. Comput. Phys. 71 (2), 231–303 (1987).
18. 18.
A. V. Rodionov, “Methods of increasing the accuracy in Godunov’s scheme,” USSR Comput. Math. Math. Phys. 27 (6), 164–169 (1987).
19. 19.
F. Alcrudoa and F. Benkhaldoun, “Exact solutions to the Riemann problem of the shallow water equations with a bottom step,” Comput. Fluids 30 (6), 643–671 (2001).
20. 20.
V. V. Ostapenko, “Dam-break flows over a bottom step,” J. Appl. Mech. Tech. Phys. 44 (4), 495–505 (2003).
21. 21.
V. V. Ostapenko, “Dam-break flows over a bottom drop,” J. Appl. Mech. Tech. Phys. 44 (6), 839–851 (2003).
22. 22.
V. V. Ostapenko and A. A. Malysheva, “Flows resulting from the incidence of a discontinuous wave on a bottom step,” J. Appl. Mech. Tech. Phys. 47 (2), 157–168 (2006).
23. 23.
V. V. Ostapenko and E. V. Shinkarenko, “Flows formed after the passage of a discontinuous wave over a bottom drop,” Fluid Mech. 44 (1), 88–102 (2009).
24. 24.
V. V. Belikov, N. M. Borisova, and V. V. Ostapenko, “Improvement of numerical methods for simulating hydraulic structures with sharp bottom level differences,” in Safety of Energy Structures (Nauchno-Isled. Inst. Energ. Sooruzh., Moscow, 2007), Vol. 16, pp. 79–89 [in Russian].Google Scholar
25. 25.
A. S. Petrosyan, Additional Topics in Free-Boundary Heavy Fluid Dynamics (Inst. Kosm. Issled. Ross. Akad. Nauk, Moscow, 2010) [in Russian].Google Scholar
26. 26.
O. V. Bulatov, “Analytical and numerical Riemann solutions of the Saint Venant equations for forward-and backward-facing step flows,” Comput. Math. Math. Phys. 52 (1), 158–171 (2014).
27. 27.
E. Han and G. Warnecke, “Exact Riemann solutions to shallow water equations,” Q. Appl. Math. 72 (3), 407–453 (2014).
28. 28.
V. V. Belikov, A. N. Militeev, and V. V. Kochetkov, “Software package for computing dam break waves (BOR),” RF Patent No. 2001610638 (2001).Google Scholar
29. 29.
V. V. Belikov and V. V. Kochetkov, “STREAM_2D software package for computing flows, bed deformations, and pollutant dispersion in open channels,” RF Patent No. 2014612181 (2014).Google Scholar
30. 30.
V. V. Belikov and S. V. Kovalev, “Numerical investigations for solution of hydraulic problems,” Power Technol. Eng. 43 (5), 296–301 (2009).Google Scholar
31. 31.
V. V. Belikov, E. S. Vasileva, and A. M. Prudovskii, “Numerical modeling of a breach wave through the dam at the Krasnodar reservoir,” Power Technol. Eng. 44 (4), 269–278 (2010).Google Scholar
32. 32.
V. V. Belikov and N. M. Borisova, “Numerical study of dam break waves,” in Safety of Energy Structures (Moscow, 2010), Vol. 17, pp. 205–214 [in Russian].Google Scholar
33. 33.
V. V. Belikov, S. V. Norin, and S. Ya. Shkol’nikov, “On dam break in polders,” Gidrotekh. Stroitel’stvo, No. 12, 25–34 (2014).Google Scholar
34. 34.
N. I. Alekseevskiy, I. N. Krylenko, V. V. Belikov, V. V. Kochetkov, and S. V. Norin, “Numerical hydrodynamic modeling of inundation in Krymsk on July 6–7, 2012,” Power Technol. Eng. 48 (3), 179–186 (2014).
35. 35.
D. R. Bazarov and A. N. Militeev, “A two-dimensional mathematical model of the horizontal deformations of river channels,” Water Resources 26 (1), 17–21 (1999).Google Scholar
36. 36.
Y. Huang, N. Zhang, and Y. Pei, “Well-balanced finite volume scheme for shallow water flooding and drying over arbitrary topography,” Eng. Appl. Comput. Fluid Mech. 7 (1), 40–54 (2013).Google Scholar
37. 37.
Q. Liang and A. G. Borthwick, “Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography,” Comput. Fluids 38 (2), 221–234 (2009).
38. 38.
L. Song, J. Zhou, Q. Li, X. Yang, and Y. Zhang, “An unstructured finite volume model for dam-break floods with wet/dry fronts over complex topography,” Int. J. Numer. Methods Fluids 67 (8), 960–980 (2011).
39. 39.
V. I. Bukreev, A. V. Gusev, A. A. Malysheva, and I. A. Malysheva, “Experimental verification of the gas-hydraulic analogy with reference to the dam-break problem,” Fluid Dyn. 39 (5), 801–809 (2004).
40. 40.
V. V. Ostapenko, “Modified shallow water equations which admit the propagation of discontinuous waves over a dry bed,” J. Appl. Mech. Tech. Phys. 48 (6), 795–812 (2007).
41. 41.
G. F. Carrier and H. P. Greenspan, “Water waves of finite amplitude on a sloping beach,” J. Fluid Mech. 4 (1), 97–109 (1958).
42. 42.
M. Kawahara and T. Umetsu, “Finite element method for moving boundary problems in river flow,” Int. J. Numer. Methods Fluids 6 (6), 365–386 (1986).

## Copyright information

© Pleiades Publishing, Ltd. 2017

## Authors and Affiliations

1. 1.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia
2. 2.Water Problems InstituteRussian Academy of SciencesMoscowRussia