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.
Accepted:
Published online:
Xinhua Lu 1; Bingjiang Dong 2; Bing Mao 3; Xiaofeng Zhang 1
@article{CRMECA_2015__343_7-8_429_0, author = {Xinhua Lu and Bingjiang Dong and Bing Mao and Xiaofeng Zhang}, title = {A robust and well-balanced numerical model for solving the two-layer shallow water equations over uneven topography}, journal = {Comptes Rendus. M\'ecanique}, pages = {429--442}, publisher = {Elsevier}, volume = {343}, number = {7-8}, year = {2015}, doi = {10.1016/j.crme.2015.05.002}, language = {en}, }
TY - JOUR AU - Xinhua Lu AU - Bingjiang Dong AU - Bing Mao AU - Xiaofeng Zhang TI - A robust and well-balanced numerical model for solving the two-layer shallow water equations over uneven topography JO - Comptes Rendus. Mécanique PY - 2015 SP - 429 EP - 442 VL - 343 IS - 7-8 PB - Elsevier DO - 10.1016/j.crme.2015.05.002 LA - en ID - CRMECA_2015__343_7-8_429_0 ER -
%0 Journal Article %A Xinhua Lu %A Bingjiang Dong %A Bing Mao %A Xiaofeng Zhang %T A robust and well-balanced numerical model for solving the two-layer shallow water equations over uneven topography %J Comptes Rendus. Mécanique %D 2015 %P 429-442 %V 343 %N 7-8 %I Elsevier %R 10.1016/j.crme.2015.05.002 %G en %F CRMECA_2015__343_7-8_429_0
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] Two-layer shallow water computation of mud flow intrusions into quiescent water, J. Hydraul. Res., Volume 45 (2007), pp. 13-25
[2] Idealized numerical simulation of breaking water wave propagating over a viscous mud layer, Phys. Fluids, Volume 24 (2012), p. 112104
[3] Two-layer hydraulics: a functional approach, J. Fluid Mech., Volume 223 (2006), pp. 135-163
[4] An Introduction to Internal Wave, Lecture Notes, Royal NIOZ, Texel, 2008
[5] Wind-induced chaotic mixing in a two-layer density-stratified shallow flow, J. Hydraul. Res., Volume 52 (2014), pp. 219-227
[6] Shock-Capturing Methods for Free-Surface Shallow Flows, Wiley, 2001
[7] Godunov-type methods for free-surface shallow flows: a review, J. Hydraul. Res., Volume 45 (2007), pp. 736-751
[8] Review of wetting and drying algorithms for numerical tidal flow models, Int. J. Numer. Methods Fluids, Volume 71 (2013), pp. 473-487
[9] Central-upwind schemes for two-layer shallow water equations, SIAM J. Sci. Comput., Volume 31 (2009), pp. 1742-1773
[10] ADER schemes on unstructured meshes for nonconservative hyperbolic systems: applications to geophysical flows, Comput. Fluids, Volume 38 (2009), pp. 1731-1748
[11] Numerical treatment of the loss of hyperbolicity of the two-layer shallow-water system, J. Sci. Comput., Volume 48 (2011), pp. 16-40
[12] Definition and weak stability of non-conservative products, J. Math. Pures Appl., Volume 74 (1995), pp. 483-548
[13] A comment on the computation of non-conservative products, J. Comput. Phys., Volume 229 (2010), pp. 2759-2763
[14] Solution properties and approximate Riemann solvers for two-layer shallow flow models, Comput. Fluids, Volume 44 (2011), pp. 202-220
[15] Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids, Volume 23 (1994), pp. 1049-1071
[16] 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] 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] 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] 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] Adaptive Q-tree Godunov-type scheme for shallow water equations, Int. J. Numer. Methods Fluids, Volume 35 (2001), pp. 247-280
[21] Adaptive quadtree simulation of shallow flows with wet–dry fronts over complex topography, Comput. Fluids, Volume 38 (2009), pp. 221-234
[22] On mathematical balancing of a two-layer shallow flow model, Edinburgh, UK (2010)
[23] A fast adaptive quadtree scheme for a two-layer shallow water model, J. Comput. Phys., Volume 230 (2011), pp. 4848-4870
[24] The hydraulics of two flowing layers with different densities, J. Fluid Mech., Volume 163 (1986), pp. 27-58
[25] Two-layer shallow water system: a relaxation approach, SIAM J. Sci. Comput., Volume 31 (2009), pp. 1603-1627
[26] Well-balanced bicharacteristic-based scheme for multilayer shallow water flows including wet/dry fronts, J. Comput. Phys., Volume 235 (2013), pp. 82-113
[27] Computational dam-break hydraulics over erodible sediment bed, J. Hydraul. Eng., Volume 130 (2004), pp. 689-703
[28] On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., Volume 25 (1983), pp. 35-61
[29] Restoration of the contact surface in the HLL-Riemann solver, Shock Waves, Volume 4 (1994), pp. 25-34
[30] Simplified second-order Godunov-type methods, SIAM J. Sci. Stat. Comput., Volume 9 (1988), pp. 455-473
[31] Unstructured mixed grid and SIMPLE algorithm based model for 2D-SWE, Proc. Eng., Volume 28 (2012), pp. 117-121
[32] Kernel analysis of the discretized finite difference and finite element shallow-water models, SIAM J. Sci. Comput., Volume 31 (2008), pp. 531-556
[33] A finite element method for solving the shallow water equations on the sphere, Ocean Model., Volume 28 (2009), pp. 12-23
[34] A numerical method for the two layer shallow water equations with dry states, Ocean Model., Volume 72 (2013), pp. 80-91
[35] A 2D numerical model of wave run-up and overtopping, Coast. Eng., Volume 47 (2002), pp. 1-26
[36] 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] Topography discretization techniques for Godunov-type shallow water numerical models: a comparative study, J. Hydraul. Res., Volume 51 (2013), pp. 351-367
[38] Improved implementation of the HLL approximate Riemann solver for one-dimensional open channel flows, J. Hydraul. Res., Volume 46 (2008), pp. 21-34
[39] Dam break flow computation based on an efficient flux vector splitting, J. Comput. Appl. Math., Volume 234 (2010), pp. 2143-2151
[40] Open boundary conditions in stratified ocean models, J. Mar. Syst., Volume 16 (1998), pp. 297-322
[41] 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] Open boundary conditions for nonlinear channel flow, Ocean Model., Volume 24 (2008), pp. 108-121
[43] Divergence form for bed slope source term in shallow water equations, J. Hydraul. Eng., Volume 132 (2006), pp. 652-665
[44] An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment, Modél. Math. Anal. Numér., Volume 42 (2008), pp. 683-698
[45] 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] 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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