Comptes Rendus
no. 7-8
A robust and well-balanced numerical model for solving the two-layer shallow water equations over uneven topography
Xinhua Lu1  ; Bingjiang Dong2  ; Bing Mao3  ; Xiaofeng Zhang1
1 State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China
2 Hydrology Bureau, Yangtze River Water Resource Commission, Wuhan 430010, China
3 Yangtze River Scientific Research Institute, Wuhan 430015, China
Comptes Rendus. Mécanique, Volume 343 (2015) no. 7-8, pp. 429-442.

A robust and well-balanced numerical model is developed for solving the two-layer shallow water equations based on the approximate Riemann solver in the framework of finite-volume methods. The HLL (Harten, Lax, and van Leer) solver is employed to calculate the numerical fluxes. The numerical balance between the flux gradient and the source terms is achieved by using a balance-reformulation method. To obtain exactly the lake-at-rest solutions as the water depth is chosen as the conserved variable for the continuity equations, a modified HLL flux formulation is proposed for mass flux calculations. Several numerical tests used to validate the performance of the developed numerical model. The results show that the developed model is accurate, well balanced, and that it predicts no oscillations around large gradients.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2015.05.002
Mots clés : Two-layer system, Well-balanced model, Nonconservative, 2LSWE, HLL
Xinhua Lu 1 ; Bingjiang Dong 2 ; Bing Mao 3 ; Xiaofeng Zhang 1

1 State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China
2 Hydrology Bureau, Yangtze River Water Resource Commission, Wuhan 430010, China
3 Yangtze River Scientific Research Institute, Wuhan 430015, China 
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Xinhua Lu; Bingjiang Dong; Bing Mao; Xiaofeng Zhang. A robust and well-balanced numerical model for solving the two-layer shallow water equations over uneven topography. Comptes Rendus. Mécanique, Volume 343 (2015) no. 7-8, pp. 429-442. doi : 10.1016/j.crme.2015.05.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2015.05.002/

[1] S.-C. Chen; S.-H. Peng; H. Capart Two-layer shallow water computation of mud flow intrusions into quiescent water, J. Hydraul. Res., Volume 45 (2007), pp. 13-25

[2] Y. Hu; X. Guo; X. Lu; Y. Liu; R.A. Dalrymple; L. Shen Idealized numerical simulation of breaking water wave propagating over a viscous mud layer, Phys. Fluids, Volume 24 (2012), p. 112104

[3] S.B. Dalziel Two-layer hydraulics: a functional approach, J. Fluid Mech., Volume 223 (2006), pp. 135-163

[4] T. Gerkema; J. Zimmerman An Introduction to Internal Wave, Lecture Notes, Royal NIOZ, Texel, 2008

[5] W.-K. Lee; A.G.L. Borthwick; P.H. Taylor Wind-induced chaotic mixing in a two-layer density-stratified shallow flow, J. Hydraul. Res., Volume 52 (2014), pp. 219-227

[6] E.F. Toro Shock-Capturing Methods for Free-Surface Shallow Flows, Wiley, 2001

[7] E.F. Toro; P. Garcia-Navarro Godunov-type methods for free-surface shallow flows: a review, J. Hydraul. Res., Volume 45 (2007), pp. 736-751

[8] S.C. Medeiros; S.C. Hagen Review of wetting and drying algorithms for numerical tidal flow models, Int. J. Numer. Methods Fluids, Volume 71 (2013), pp. 473-487

[9] A. Kurganov; G. Petrova Central-upwind schemes for two-layer shallow water equations, SIAM J. Sci. Comput., Volume 31 (2009), pp. 1742-1773

[10] M. Dumbser; M. Castro; C. Parés; E.F. Toro ADER schemes on unstructured meshes for nonconservative hyperbolic systems: applications to geophysical flows, Comput. Fluids, Volume 38 (2009), pp. 1731-1748

[11] M.J. Castro-Díaz; E.D. Fernández-Nieto; J.M. González-Vida; C. Parés-Madroñal Numerical treatment of the loss of hyperbolicity of the two-layer shallow-water system, J. Sci. Comput., Volume 48 (2011), pp. 16-40

[12] G. Dal Maso; P. LeFloch; F. Murat Definition and weak stability of non-conservative products, J. Math. Pures Appl., Volume 74 (1995), pp. 483-548

[13] R. Abgrall; S. Karni A comment on the computation of non-conservative products, J. Comput. Phys., Volume 229 (2010), pp. 2759-2763

[14] B. Spinewine; V. Guinot; S. Soares-Frazão; Y. Zech Solution properties and approximate Riemann solvers for two-layer shallow flow models, Comput. Fluids, Volume 44 (2011), pp. 202-220

[15] A. Bermudez; M. Vazquez Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids, Volume 23 (1994), pp. 1049-1071

[16] S. Bryson; Y. Epshteyn; A. Kurganov; G. Petrova Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system, Modél. Math. Anal. Numér., Volume 45 (2011), pp. 423-446

[17] M. Kazolea; A.I. Delis A well-balanced shock-capturing hybrid finite volume–finite difference numerical scheme for extended 1D Boussinesq models, Appl. Numer. Math., Volume 67 (2013), pp. 167-186

[18] J.G. Zhou; D.M. Causon; C.G. Mingham; D.M. Ingram The surface gradient method for the treatment of source terms in the shallow-water equations, J. Comput. Phys., Volume 168 (2001), pp. 1-25

[19] E. Audusse; F. Bouchut; M.-O. Bristeau; R. Klein; B. Perthame A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., Volume 25 (2004), pp. 2050-2065

[20] B. Rogers; M. Fujihara; A.G.L. Borthwick Adaptive Q-tree Godunov-type scheme for shallow water equations, Int. J. Numer. Methods Fluids, Volume 35 (2001), pp. 247-280

[21] Q. Liang; A.G.L. Borthwick Adaptive quadtree simulation of shallow flows with wet–dry fronts over complex topography, Comput. Fluids, Volume 38 (2009), pp. 221-234

[22] W.-K. Lee; A.G.L. Borthwick; P.H. Taylor On mathematical balancing of a two-layer shallow flow model, Edinburgh, UK (2010)

[23] W.-K. Lee; A.G. Borthwick; P.H. Taylor A fast adaptive quadtree scheme for a two-layer shallow water model, J. Comput. Phys., Volume 230 (2011), pp. 4848-4870

[24] L. Arm The hydraulics of two flowing layers with different densities, J. Fluid Mech., Volume 163 (1986), pp. 27-58

[25] R. Abgrall; S. Karni Two-layer shallow water system: a relaxation approach, SIAM J. Sci. Comput., Volume 31 (2009), pp. 1603-1627

[26] M. Dudzinski; M. Lukáčová-Medvid'ová Well-balanced bicharacteristic-based scheme for multilayer shallow water flows including wet/dry fronts, J. Comput. Phys., Volume 235 (2013), pp. 82-113

[27] Z. Cao; G. Pender; S. Wallis; P. Carling Computational dam-break hydraulics over erodible sediment bed, J. Hydraul. Eng., Volume 130 (2004), pp. 689-703

[28] A. Harten; P.D. Lax; B. van Leer On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., Volume 25 (1983), pp. 35-61

[29] E.F. Toro; M. Spruce; W. Speares Restoration of the contact surface in the HLL-Riemann solver, Shock Waves, Volume 4 (1994), pp. 25-34

[30] S. Davis Simplified second-order Godunov-type methods, SIAM J. Sci. Stat. Comput., Volume 9 (1988), pp. 455-473

[31] X. Lu; B. Dong; B. Mao; X. Zhang Unstructured mixed grid and SIMPLE algorithm based model for 2D-SWE, Proc. Eng., Volume 28 (2012), pp. 117-121

[32] V. Rostand; D.Y.L. Roux; G. Carey Kernel analysis of the discretized finite difference and finite element shallow-water models, SIAM J. Sci. Comput., Volume 31 (2008), pp. 531-556

[33] R. Comblen; S. Legrand; E. Deleersnijder; V. Legat A finite element method for solving the shallow water equations on the sphere, Ocean Model., Volume 28 (2009), pp. 12-23

[34] K.T. Mandli A numerical method for the two layer shallow water equations with dry states, Ocean Model., Volume 72 (2013), pp. 80-91

[35] M.E. Hubbard; N. Dodd A 2D numerical model of wave run-up and overtopping, Coast. Eng., Volume 47 (2002), pp. 1-26

[36] W. Wu; R. Marsooli A depth-averaged 2D shallow water model for breaking and non-breaking long waves affected by rigid vegetation, J. Hydraul. Res., Volume 50 (2012), pp. 558-575

[37] G. Kesserwani Topography discretization techniques for Godunov-type shallow water numerical models: a comparative study, J. Hydraul. Res., Volume 51 (2013), pp. 351-367

[38] X. Ying; S.S.Y. Wang Improved implementation of the HLL approximate Riemann solver for one-dimensional open channel flows, J. Hydraul. Res., Volume 46 (2008), pp. 21-34

[39] S. Erpicum; B. Dewals; P. Archambeau; M. Pirotton Dam break flow computation based on an efficient flux vector splitting, J. Comput. Appl. Math., Volume 234 (2010), pp. 2143-2151

[40] T.G. Jensen Open boundary conditions in stratified ocean models, J. Mar. Syst., Volume 16 (1998), pp. 297-322

[41] B.F. Sanders High-resolution and non-oscillatory solution of the St. Venant equations in non-rectangular and non-prismatic channels, J. Hydraul. Res., Volume 39 (2001), pp. 321-330

[42] J. Nycander; A.M. Hogg; L.M. Frankcombe Open boundary conditions for nonlinear channel flow, Ocean Model., Volume 24 (2008), pp. 108-121

[43] A. Valiani; L. Begnudelli Divergence form for bed slope source term in shallow water equations, J. Hydraul. Eng., Volume 132 (2006), pp. 652-665

[44] F. Bouchut; T.M. de Luna An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment, Modél. Math. Anal. Numér., Volume 42 (2008), pp. 683-698

[45] M. Castro; J. Macías; C. Parés A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system, Modél. Math. Anal. Numér., Volume 35 (2001), pp. 107-127

[46] F. Bouchut; V. Zeitlin A robust well-balanced scheme for multi-layer shallow water equations, Discrete Contin. Dyn. Syst., Ser. B, Volume 13 (2010), pp. 739-758

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