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A robust second-order surface reconstruction for shallow water flows with a discontinuous topography and a Manning friction. (English) Zbl 1433.76104
Summary: A second-order surface reconstruction (SR) method for the shallow water equations with a discontinuous bottom topography and a Manning friction source term is presented. We redefine the water surface level at the cell interface by using the minimum difference between the bottom level and the original water surface level. The reconstructed water surface level is used to define the intermediate bottom level and the intermediate water height at the cell interface. We propose an explicit-implicit method to address the friction source term. The new second-order SR scheme together with the explicit-implicit method can preserve a special steady-state solution of the system and can maintain the positivity of the water depth. We also extend the new scheme to two-dimensional shallow water flows. To demonstrate the robustness and effectiveness of the new scheme, we use several classical numerical experiments for the shallow water flows over a complex bottom topography.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
35L65 Hyperbolic conservation laws
35Q35 PDEs in connection with fluid mechanics
Software:
HLLE
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[1] Saint-Venant: Théorie du mouvement non-permanent des eaux avec application aux crues des rivières et à i’introduction des mares dans leur lit., Comptes Rendus Hebdomadaires Des Séances De Lacadémie Des Sciences 73 · JFM 03.0482.04
[2] Bollermann, A.; Chen, G.; Kurganov, A.; Noelle, S., A well-balanced reconstruction of wet/dry fronts for the shallow water equations, J. Sci. Comput., 56, 2, 267-290 (2013) · Zbl 1426.76337
[3] Liang, Q.; Marche, F., Numerical resolution of well-balanced shallow water equations with complex source terms, Adv. Water Resour., 32, 6, 873-884 (2009)
[4] Chertock, A.; Cui, S.; Kurganov, A.; Wu, T., Well-balanced positivity preserving central-upwind scheme for the shallow water system with friction terms, Int. J. Numer. Methods Fluids, 78, 6, 355-383 (2015)
[5] Michel-Dansac, V.; Berthon, C.; Clain, S.; Foucher, F., A well-balanced scheme for the shallow-water equations with topography or manning friction, J. Comput. Phys., 335, 115-154 (2017) · Zbl 1375.35389
[6] Chertock, A.; Cui, S.; Kurganov, A.; Wu, T., Steady state and sign preserving semi-implicit runge-kutta methods for odes with stiff damping term, SIAM J. Numer. Anal., 53, 4, 2008-2029 (2015) · Zbl 1327.65128
[7] Dong, J., Li, D.F.: A reliable second-order hydrostatic reconstruction for shallow water flows with the friction term and the bed source term, Journal of Computational and Applied Mathematics. In: Revision
[8] Gosse, L.; Leroux, A-Y, A well-balanced scheme designed for inhomogeneous scalar conservation laws, Comptes Rendus De L Academie Des Sciences Serie I-mathematique, 323, 5, 543-546 (1996) · Zbl 0858.65091
[9] Greenberg, JM; LeRoux, A-Y, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33, 1, 1-16 (1996)
[10] Kurganov, A.; Petrova, G., A second-order well-balanced positivity preserving central-upwind scheme for the saint-venant system, Commun. Math. Sci., 5, 1, 133-160 (2007) · Zbl 1226.76008
[11] Jin, S., A Steady-State Capturing Method For Hyperbolic Systems With Geometrical Source Terms (2004), New York: Springer, New York · Zbl 1045.35002
[12] Audusse, E.; Bouchut, F.; Bristeau, MO; Klein, R.; Perthame, B., A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, Siam J. Sci. Comput., 25, 6, 2050-2065 (2004) · Zbl 1133.65308
[13] Bollermann, A.; Noelle, S.; Lukáčová-medvidová, M., Finite volume evolution galerkin methods for the shallow water equations with dry beds, Commun. Comput. Phys., 10, 2, 371-404 (2011) · Zbl 1364.76109
[14] Xing, Y.; Shu, CW, A new approach of high order well-balanced finite volume weno schemes and discontinuous galerkin methods for a class of hyperbolic systems with source, Commun. Comput. Phys., 1, 1, 567-598 (2006) · Zbl 1089.65091
[15] Noelle, S.; Pankratz, N.; Puppo, G.; Natvig, JR, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comput. Phys., 213, 2, 474-499 (2006) · Zbl 1088.76037
[16] Xing, Y.; Zhang, X.; Shu, CW, Positivity-preserving high order well-balanced discontinuous galerkin methods for the shallow water equations✩✩✩, Adv. Water Resour., 33, 12, 1476-1493 (2010)
[17] Chen, G.; Noelle, S., A new hydrostatic reconstruction scheme based on subcell reconstructions, Siam J. Numer. Anal., 55, 2, 758-784 (2017) · Zbl 1365.76146
[18] Sampson, J.; Easton, A.; Singh, M., Moving boundary shallow water flow above parabolic bottom topography, Anziam J., 47, 373-387 (2006)
[19] Buttinger-Kreuzhuber, A.; Horváth, Z.; Noelle, S.; Blöschl, G.; Waser, J., A new second-order shallow water scheme on two-dimensional structured grids over abrupt topography, Adv. Water Resour., 127, 89-108 (2019)
[20] de Luna, TM; Díaz, MJC; Parés, C., Reliability of first order numerical schemes for solving shallow water system over abrupt topography, Appl. Math. Comput., 219, 17, 9012-9032 (2013) · Zbl 1290.76089
[21] Dong, J., Li, D.F., Chen, G.: A new second-order modified hydrostatic reconstruction for the shallow water flows with a discontinuous topography, Applied Numerical Mathematics, In revision
[22] Dong, J., Li, D.F.: An efficient second-order modified hydrostatic reconstruction for the two-dimensional shallow water flows over complex topography, COMPUTERS & FLUIDS Under review
[23] Dong, J.; Li, DF, An effect non-staggered central scheme based on new hydrostatic reconstruction, Appl. Math. Comput., 372, 124992 (2020) · Zbl 1433.76105
[24] Xia, X.; Liang, Q.; Ming, X.; Hou, J., An efficient and stable hydrodynamic model with novel source term discretization schemes for overland flow and flood simulations, Water Resour. Res., 53, 5, 3730-3759 (2017)
[25] Lie, KA; Noelle, S., On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws, Siam J. Sci. Comput., 24, 4, 1157-1174 (2003) · Zbl 1038.65078
[26] Sweby, PK, High resolution schemes using flux limiters for hyperbolic conservation laws, Siam J. Numer. Anal., 21, 5, 995-1011 (1984) · Zbl 0565.65048
[27] Jin, S.; Wen, X., Two interface-type numerical methods for computing hyperbolic systems with geometrical source terms having concentrations, Siam J. Sci. Comput., 26, 6, 2079-2101 (2006) · Zbl 1083.35062
[28] Vanleer, B., Towards the ultimate conservative difference scheme v second order sequel to godunov’s method, J. Comput. Phys., 32, 1, 101-136 (1979) · Zbl 1364.65223
[29] Harten, A.; Lax, PD; Leer, BV, On upstream differencing and godunov-type schemes for hyperbolic conservation laws, Siam Rev., 25, 1, 35-61 (1983) · Zbl 0565.65051
[30] Shu, CW; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 2, 439-471 (1989)
[31] Chertock, A.; Cui, S.; Kurganov, A.; Wu, T., Well-balanced positivity preserving central-upwind scheme for the shallow water system with friction terms, Int. J. Numer. Methods Fluids, 78, 6, 355-383 (2015)
[32] Soares-FrazãO, S., Experiments of dam-break wave over a triangular bottom sill, J. Hydraul. Res., 45, sup1, 19-26 (2007)
[33] Joe Sampson, M.S., Easton, A.: Moving boundary shallow water flow in parabolic bottom topography, Anziam Journal. · Zbl 1347.76010
[34] Delestre, O.; Cordier, S.; Darboux, F.; James, F., A limitation of the hydrostatic reconstruction technique for shallow water equations, Comptes Rendus Mathematique, 350, 13-14, 677-681 (2012) · Zbl 1251.35081
[35] Kawahara, M.; Umetsu, T., Finite element method for moving boundary problems in river flow, Int. J. Numer. Methods Fluids, 6, 6, 365-386 (2010) · Zbl 0597.76014
[36] 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, Int. J. Numer. Methods Fluids, 67, 8, 960-980 (2011) · Zbl 1316.76060
[37] Briggs, MJ; Synolakis, CE; Harkins, GS; Green, DR, Laboratory experiments of tsunami runup on a circular island, Pure Appl. Geophys., 144, 3-4, 569-593 (1995)
[38] Liu, X.; Albright, J.; Epshteyn, Y.; Kurganov, A., Well-balanced positivity preserving central-upwind scheme with a novel wet/dry reconstruction on triangular grids for the saint-venant system, J. Comput. Phys., 374, 213-236 (2018) · Zbl 1416.65295
[39] Nikolos, IK; Delis, AI, An unstructured node-centered finite volume scheme for shallow water flows with wet/dry fronts over complex topography, Comput. Methods Appl. Mechan. Eng., 198, 47, 3723-3750 (2009) · Zbl 1230.76035
[40] Bradford, SF; Sanders, BF, Finite-volume model for shallow-water flooding of arbitrary topography, J. Hydraul. Eng., 128, 3, 289-298 (2002)
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