Simulation of shallow water flows with shoaling areas and bottom discontinuities

A. I. Aleksyuk   ; V. V. Belikov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 2, pp. 316-338

Résumé

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.

Bibliographie

[1] Stoker Dzh. Dzh., Volny na vode. Matematicheskaya teoriya i prilozheniya, Inostr. lit., M., 1959

[2] Voltsinger N. E., Pyaskovskii R. V., Teoriya melkoi vody. Okeanologicheskie zadachi i chislennye metody, Gidrometeoizdat, L., 1977

[3] Kulikovskii A. G., Pogorelov N. V., Semenov A. Yu., Matematicheskie voprosy chislennogo resheniya giperbolicheskikh sistem uravnenii, Fizmatlit, M., 2001

[4] Garcia F. R., Kahawita R., “Numerical solution of the shallow water equations with a maccormack type finite difference scheme”, Proc. 4th Internat. Conference on Math. Modelling in Science and Technology (15–17 Aug., 1983), 669–673 | MR

[5] Delis A. I., Skeels C. P., Ryrie S. C., “Evaluation of some approximate Riemann solvers for transient open channel flows”, J. of Hydraulic Research, 38:3 (2000), 217–231 | DOI

[6] Belikov V. V., Semenov A. Yu., Tr. X konf. molodykh uchenykh Mosk. fiz.-tekhn. in-ta, Dep. v VINITI Ch. 1. No 5983-85 Dep. (23 marta–7 aprelya 1985), 179–214

[7] Belikov V. V., Semenov A. Yu., “Primenenie metoda Godunova s modifikatsiei Kolgana k raschetu planov techenii v nizhnikh befakh vodopropusknykh trub”, Gidravlika dorozhnykh vodopropusknykh sooruzhenii, SPI, Saratov, 1985, 54–57

[8] Belikov V. V., “Beziteratsionnyi algoritm rascheta raspada razryva dlya uravnenii melkoi vody”, Puti povysheniya proizvoditelnosti truda, sokrascheniya srokov proektirovaniya i stroitelstva transportnykh sooruzhenii, VNII transp. str-va (TsNIIS), M., 1986, 81–85

[9] Belikov V. V., Chislennoe modelirovanie techenii zhidkosti so svobodnoi poverkhnostyu i deformiruemym dnom, Diss. ... k. fiz.-matem. nauk, M., 1987, 132 pp.

[10] Belikov V. V., Semenov A. Yu., Yavnyi chislennyi metod raspada razryvov dlya resheniya uravnenii melkoi vody, Preprint No 42, In-t obschei fiz. AN SSSR, M., 1988, 44 pp.

[11] Belikov V. V., Semenov A. Yu., “Chislennyi metod raspada razryva dlya resheniya uravnenii teorii melkoi vody”, Zh. vychisl. matem. i matem. fiz., 37:8 (1997), 1006–1019 | Zbl

[12] Belikov V. V., Semenov A. Yu., “Postroenie chislennykh metodov raspada razryva dlya resheniya uravnenii teorii melkoi vody”, Vychislitelnaya gidrodinamika prirodnykh techenii, Tr. IOFAN, 53, Fizmatlit, M., 1997

[13] Belikov V. V., Semenov A. Yu., “A Godunov's type method Based on an exact solution to the Riemann problem for the shallow-water equations”, Proc. 4 Eur. Comp. Fluid Dyn. Conf., ECCOMAS 98, Part 1, v. 1, WILEY, New York, 1998, 310–315

[14] Papa L., “Application of the courant-isaacson-rees method to solve the shallow-water hydrodynamic equations”, Appl. Math. and Comput., 15:1 (1984), 85–92 | MR | Zbl

[15] Glaister P., “A weak formulation of Roe's approximate Riemann solver applied to the St. Venant equations”, J. Comput. Phys., 116:1 (1995), 189–191 | DOI | MR | Zbl

[16] Van Leer B., “Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method”, J. Comput. Phys., 32:1 (1979), 101–136 | DOI | MR

[17] Harten A., Engquist B., Osher S., Chakravarthy S. B., “Uniformly high order accurate non-oscillatory shemes, III”, J. Comput. Phys., 71:2 (1987), 231–303 | DOI | MR | Zbl

[18] Rodionov A. V., “Povyshenie poryadka approksimatsii skhemy S. K. Godunova”, Zh. vychisl. matem. i matem. fiz., 27:12 (1987), 1853–1860 | Zbl

[19] Alcrudoa F., Benkhaldoun F., “Exact solutions to the Riemann problem of the shallow water equations with a bottom step”, Comput. Fluids, 30:6 (2001), 643–671 | DOI | MR

[20] Ostapenko V. V., “Techeniya, voznikayuschie pri razrushenii plotiny nad stupenkoi dna”, PMTF, 44:4 (2003), 51–63

[21] Ostapenko V. V., “Techeniya, voznikayuschie pri razrushenii plotiny nad ustupom dna”, PMTF, 44:6 (2003), 107–122 | Zbl

[22] Ostapenko V. V., Malysheva A. A., “Techeniya, voznikayuschie pri nabeganii preryvnoi volny na stupenku dna”, Prikl. matem. i tekhn. fiz., 47:2 (2006), 8–22

[23] Ostapenko V. V., Shinkarenko E. V., “Techeniya, voznikayuschie posle prokhozhdeniya preryvnoi volny nad ustupom dna”, Izv. RAN. Mekhan. zhidkosti i gaza, 2009, no. 1, 106–122

[24] Belikov V. V., Borisova N. M., Ostapenko V. V., “Sovershenstvovanie metodov chislennogo modelirovaniya gidrotekhnicheskikh sooruzhenii s rezkimi perepadami otmetok dna”, Bezopasnost energeticheskikh sooruzhenii, 16, OAO “NIIES”, M., 2007, 79–89

[25] Petrosyan A. S., Dopolnitelnye glavy gidrodinamiki tyazheloi zhidkosti so svobodnoi granitsei, Mekhanika, upravlenie, informatika, IKI RAN, M., 2010

[26] Bulatov O. V., “Analiticheskie i chislennye resheniya uravnenii Sen-Venana dlya nekotorykh zadach o raspade razryva nad ustupom i stupenkoi dna”, Zh. vychisl. matem. i matem. fiz., 52:1 (2014), 150–164

[27] Han E., Warnecke G., “Exact Riemann solutions to shallow water equations”, Quart. Appl. Math., 72:3 (2014), 407–453 | DOI | MR | Zbl

[28] Belikov V. V., Militeev A. N., Kochetkov V. V., Kompleks programm dlya rascheta voln proryva (BOR), Svidetelstvo ob ofitsialnoi registratsii programmy dlya EVM No 2001610638, Rossiiskoe agentstvo po patentnym i tovarnym znakam, M., 2001

[29] Belikov V. V., Kochetkov V. V., Programmnyi kompleks STREAM_2D dlya rascheta techenii, deformatsii dna i perenosa zagryaznenii v otkrytykh potokakh, Svidetelstvo ob ofitsialnoi registratsii programmy dlya EVM No 2014612181, M., 2014

[30] Belikov V. V., Kovalev S. V., “Numerical investigations for solution of hydraulic problems”, Power Technology and Engng., 43:5 (2009), 296–301

[31] Belikov V. V., Vasileva E. S., Prudovskii A. M., “Numerical modeling of a breach wave through the dam at the Krasnodar reservoir”, Power Technology and Engng., 44:4 (2010), 269–278

[32] Belikov V. V., Borisova N. M., “Chislennye issledovaniya voln proryva sudokhodnykh gidrotekhnicheskikh sooruzhenii”, Nauchno-tekhn. i proizvodstvennyi sb., Bezopasnost energeticheskikh sooruzhenii, 17, M., 2010, 205–214

[33] Belikov V. V., Norin S. V., Shkolnikov S. Ya., “O proryve damb polderov”, Gidrotekhn. stroitelstvo, 2014, no. 12, 25–34

[34] Alekseevskiy N. I., Krylenko I. N., Belikov V. V., Kochetkov V. V., Norin S. V., “Numerical hydrodynamic modeling of inundation in Krymsk on 6–7 July 2012”, Power Technology and Engng., 48:3 (2014), 179–186 | DOI

[35] Bazarov D. R., Militeev A. N., “Dvukhmernye (v plane) uravneniya dlya potokov s razmyvaemym dnom”, Vodnye resursy, 26:1 (1999), 22–26

[36] Huang Y., Zhang N., Pei Y., “Well-balanced finite volume scheme for shallow water flooding and drying over arbitrary topography”, Engng Applicat. Comput. Fluid Mech., 7:1 (2013), 40–54 | DOI

[37] Liang Q., Borthwick A. G., “Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography”, Comput. and Fluids, 38:2 (2009), 221–234 | DOI | MR | Zbl

[38] Song L., Zhou J., Li Q., Yang X., Zhang Y., “An unstructured finite volume model for dam-break floods with wet/dry fronts over complex topography”, Internat. J. Numerical Methods in Fluids, 67:8 (2011), 960–980 | DOI | MR | Zbl

[39] Bukreev V. I., Gusev A. V., Malysheva A. A., Malysheva I. A., “Eksperimentalnaya proverka gazo-gidravlicheskoi analogii na primere zadachi o razrushenii plotiny”, Izv. RAN. Mekhan. zhidkosti i gaza, 2004, no. 5, 143–152

[40] Ostapenko V. V., “Modifitsirovannye uravneniya teorii melkoi vody, dopuskayuschie rasprostranenie preryvnykh voln po sukhomu ruslu”, Prikl. matem. i tekhn. fiz., 48:6 (2007), 22–43 | Zbl

[41] Carrier G. F., Greenspan H. P., “Water waves of finite amplitude on a sloping beach”, J. Fluid Mech., 4:1 (1958), 97–109 | DOI | MR | Zbl

[42] Kawahara M., Umetsu T., “Finite element method for moving boundary problems in river flow”, Internat. J. Numerical Methods in Fluids, 6:6 (1986), 365–386 | DOI | Zbl