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Generalized finite difference method for two-dimensional shallow water equations. (English) Zbl 1403.76133
Summary: A novel meshless numerical scheme, based on the generalized finite difference method (GFDM), is proposed to accurately analyze the two-dimensional shallow water equations (SWEs). The SWEs are a hyperbolic system of first-order nonlinear partial differential equations and can be used to describe various problems in hydraulic and ocean engineering, so it is of great importance to develop an efficient and accurate numerical model to analyze the SWEs. According to split-coefficient matrix methods, the SWEs can be transformed to a characteristic form, which can easily present information of characteristic in the correct directions. The GFDM and the second-order Runge-Kutta method are adopted for spatial and temporal discretization of the characteristic form of the SWEs, respectively. The GFDM is one of the newly-developed domain-type meshless methods, so the time-consuming tasks of mesh generation and numerical quadrature can be truly avoided. To use the moving-least squares method of the GFDM, the spatial derivatives at every node can be expressed as linear combinations of nearby function values with different weighting coefficients. In order to properly cooperate with the split-coefficient matrix methods and the characteristic of the SWEs, a new way to determine the shape of star in the GFDM is proposed in this paper to capture the wave transmission. Numerical results and comparisons from several examples are provided to verify the merits of the proposed meshless scheme. Besides, the numerical results are compared with other solutions to validate the accuracy and the consistency of the proposed meshless numerical scheme.

##### MSC:
 76M20 Finite difference methods applied to problems in fluid mechanics 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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##### References:
 [1] Liggett, J. A., Fluid mechanics, (1994), McGraw-Hill Singapore [2] Hanif Chaudhry, M., Open-Channel Flow, (2008), Springer New York · Zbl 1149.76001 [3] Fennema, R. J.; Hanif Chaudhry, M., Explicit methods for 2-D transient free-surface flows, ASCE J Hydraul Eng, 116, 8, 1013-1034, (1990) [4] Molls, T.; Hanif Chaudhry, M., Depth-averaged open-channel flow model, ASCE J Hydraul Eng, 121, 6, 453-465, (1995) [5] Yoon, T. H.; Kang, S. K., Finite volume model for two-dimensional shallow water flows on unstructured grids, ASCE J Hydraul Eng, 130, 7, 678-688, (2004) [6] Kuiry, S. N.; Pramanik, K.; Sen, D., Finite volume model for shallow water equations with improved treatment of source terms, ASCE J Hydraul Eng, 134, 2, 231-242, (2008) [7] Liang, S. J.; Tang, J. H.; Wu, M. S., Solution of shallow-water equations using least-squares finite-element method, Acta Mech Sin, 24, 5, 523-532, (2008) · Zbl 1257.76051 [8] Hon, Y. C.; Cheung, K. F.; Mao, X. Z.; Kansa, E. J., Multiquadric solution for shallow water equations, ASCE J Hydraul Eng, 125, 5, 524-533, (1999) [9] Wong, S. M.; Hon, Y. C.; Golberg, M. A., Compactly supported radial basis functions for shallow water equations, Appl Math Comput, 127, 79-101, (2002) · Zbl 1126.76352 [10] Sun, C. P.; Young, D. L.; Shen, L. H.; Chen, T. F.; Hsian, C. C., Application of localized meshless methods to 2D shallow water equation problems, Eng Anal Bound Elem, 37, 1339-1350, (2013) · Zbl 1287.76184 [11] Moretti, G., The λ-scheme, Comput Fluids, 7, 3, 191-205, (1979) · Zbl 0419.76034 [12] Gabutti, B., On two upwind finite-difference schemes for hyperbolic equations in non-conservative form, Comput Fluids, 11, 3, 207-230, (1983) · Zbl 0529.65057 [13] Fennema, R. J.; Hanif Chaudhry, M., Explicit numerical schemes for unsteady free-surface flows with shocks, Water Resour Res, 22, 13, 1923-1930, (1986) [14] Fan, C. M.; Li, P. W., Numerical solutions of direct and inverse Stokes problems by the method of fundamental solutions and the Laplacian decomposition, Numer Heat Transf, Part B: Fundam, 68, 204-223, (2015) [15] Zheng, H.; Li, X., Application of the method of fundamental solutions to 2D and 3D Signorini problems, Eng Anal Bound Elem, 58, 48-57, (2015) · Zbl 1403.74335 [16] Gaspar, C., A regularized multi-level technique for solving potential problems by the method of fundamental solutions, Eng Anal Bound Elem, 57, 66-71, (2015) · Zbl 1403.65253 [17] Fan, C. M.; Chan, H. F.; Kuo, C. L.; Yeih, W., Numerical solutions of boundary detection problems using modified collocation Trefftz method and exponentially convergent scalar homotopy algorithm, Eng Anal Bound Elem, 36, 2-8, (2012) · Zbl 1259.65172 [18] Fan, C. M.; Li, H. H.; Hsu, C. Y.; Lin, C. H., Solving inverse Stokes problems by modified collocation Trefftz method, J Comput Appl Math, 268, 68-81, (2014) · Zbl 1293.65145 [19] Gu, Y.; Chen, W.; Fu, Z. J.; Zhang, B., The singular boundary method: mathematical background and application in orthotropic elastic problems, Eng Anal Bound Elem, 44, 152-160, (2014) · Zbl 1297.74145 [20] Chen, W.; Wang, F., Singular boundary method using time-dependent fundamental solutions for transient diffusion problems, Eng Anal Bound Elem, 68, 115-123, (2016) · Zbl 1403.65056 [21] Benito, J. J.; Urena, F.; Gavete, L., Influence of several factors in the generalized finite difference method, Appl Math Model, 25, 1039-1053, (2001) · Zbl 0994.65111 [22] Gavete, L.; Gavete, M. L.; Benito, J. J., Improvements of generalized finite difference method and comparison with other meshless method, Appl Math Model, 27, 831-847, (2003) · Zbl 1046.65085 [23] Benito, J. J.; Urena, F.; Gavete, L., Solving parabolic and hyperbolic equations by the generalized finite difference method, J Comput Appl Math, 209, 208-233, (2007) · Zbl 1139.35007 [24] Chan, H. F.; Fan, C. M.; Kuo, C. W., Generalized finite difference method for solving two-dimensional nonlinear obstacle problems, Eng Anal Bound Elem, 37, 1189-1196, (2013) · Zbl 1287.74056 [25] Fan, C. M.; Huang, Y. K.; Li, P. W.; Chiu, C. L., Application of the generalized finite-difference method to inverse biharmonic boundary-value problems, Numer Heat Transf, Part B: Fundam, 65, 129-154, (2014) [26] Fan, C. M.; Li, P. W.; Yeih, W., Generalized finite difference method for solving two-dimensional inverse Cauchy problems, Inverse Probl Sci Eng, 23, 5, 737-759, (2015) · Zbl 1329.65257 [27] Li, P. W.; Fan, C. M.; Chen, C. Y.; Ku, C. Y., Generalized finite difference method for numerical solutions of density-driven groundwater flows, CMES: Comput Model Eng Sci, 101, 5, 319-350, (2014) · Zbl 1356.76204 [28] Zhang, T.; Ren, Y. F.; Fan, C. M.; Li, P. W., Simulation of two-dimensional sloshing phenomenon by generalized finite difference method, Eng Anal Bound Elem, 63, 82-91, (2016) · Zbl 1403.65044 [29] Chan, H. F.; Fan, C. M., The local radial basis function collocation method for solving two-dimensional inverse Cauchy problems, Numer Heat Transf Part B: Fundam, 63, 284-303, (2013) [30] Fan, C. M.; Chien, C. S.; Chan, H. F.; Chiu, C. L., The local RBF collocation method for solving the double-diffusive natural convection in fluid-saturated porous media, Int J Heat Mass Transf, 57, 500-503, (2013) [31] Chou, C. K.; Sun, C. P.; Young, D. L.; Sladek, J.; Sladek, V., Extrapolated local radial basis functions collocation method for shallow water problems, Eng Anal Bound Elem, 50, 275-290, (2015) · Zbl 1403.76138 [32] Wu, C.; Huang, G.; Zheng, Y., Theoretical solution of dam-break shock wave, ASCE J Hydraul Eng, 125, 11, 1210-1215, (1999) [33] Valiani, A.; Begnudelli, L., Divergence form for bed slope source term in shallow water equations, ASCE J Hydraul Eng, 132, 652-665, (2006) [34] Rozovskii IL. Flow of Water in Bends of Open Channels. Israel Program for Scientific Translation, Jerusalem, Israel; 1957.
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