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The use of proper orthogonal decomposition (POD) meshless RBF-FD technique to simulate the shallow water equations. (English) Zbl 1380.65301
Summary: The main aim of this paper is to develop a fast and efficient local meshless method for solving shallow water equations in one- and two-dimensional cases. The mentioned equation has been classified in category of advection equations. The solutions of advection equations have some shock, thus, especial numerical methods should be employed for example discontinuous Galerkin and finite volume methods. Here, based on the proper orthogonal decomposition approach we want to construct a fast meshless method. To this end, we consider shallow water models and obtain a suitable time-discrete scheme based on the predictor-corrector technique. Then, by applying the proper orthogonal decomposition technique, a new set of basis functions can be built for the solution space in which the size of new solution space is less than the original problem. Thus, by employing the new bases the CPU time will be reduced. Some examples have been studied to show the efficiency of the present numerical technique.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65D05 Numerical interpolation
76M25 Other numerical methods (fluid mechanics) (MSC2010)
35Q35 PDEs in connection with fluid mechanics
Software:
Matlab
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References:
[1] Alcrudoa, F.; Benkhaldoun, F., Exact solutions to the Riemann problem of the shallow water equations with a bottom step, Comput. Fluids, 30, 643-671, (2001) · Zbl 1048.76008
[2] Benkhaldoun, F.; Halassi, A.; Ouazar, D.; Seaid, M.; Taik, Ahmed, A stabilized meshless method for time-dependent convection-dominated flow problems, Math. Comput. Simul., 137, 159-176, (2017)
[3] Benkhaldoun, F.; Seaïd, M., A simple finite volume method for the shallow water equations, J. Comput. Appl. Math., 234, 58-72, (2010) · Zbl 1273.76287
[4] Benkhaldoun, F.; Halassi, A.; Ouazar, D.; Seaid, M.; Taik, A., Slope limiters for radial basis functions applied to conservation laws with discontinuous flux function, Eng. Anal. Bound. Elem., 66, 49-65, (2016) · Zbl 1403.76137
[5] Benkhaldoun, F.; Sari, S.; Seaid, M., Projection finite volume method for shallow water flows, Math. Comput. Simul., 118, 87-101, (2015)
[6] Benkhaldoun, F.; Elmahi, I.; Moumna, A.; Seaid, M., A non-homogeneous Riemann solver for shallow water equations in porous media, Appl. Anal., 95, 2181-2202, (2016) · Zbl 1388.76164
[7] Berkooz, G.; Holmes, P.; Lumley, J. L., The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech., 25, 1, 539-575, (1993)
[8] Bermudez, A.; Vazquez-Cendon, M. E., Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids, 23, 1049-1071, (1994) · Zbl 0816.76052
[9] Berthon, C.; Foucher, F., Efficient well balanced hydrostatic upwind schemes for shallow water equations, J. Comput. Phys., 231, 4993-5015, (2012) · Zbl 1351.76095
[10] Berthon, C.; LeFloch, P.; Turpault, R., Late-time/stiff-relaxation asymptotic-preserving approximations of hyperbolic equations, Math. Comput., 82, 831-860, (2013) · Zbl 1317.65182
[11] Berthon, C.; Marche, F.; Turpault, R., An efficient scheme on wet/dry transitions for shallow water equations with friction, Comput. Fluids, 48, 192-201, (2011) · Zbl 1271.76178
[12] Berthon, C.; Boutin, B.; Turpault, R., Shock profiles for the shallow-water exner models, Adv. Appl. Math. Mech., 7, 267-294, (2015)
[13] Bistrian, D. A.; Navon, I. M., An improved algorithm for the shallow water equations model reduction: dynamic mode decomposition vs POD, Int. J. Numer. Methods Fluids, 78, 552-580, (2015)
[14] Bollig, E. F.; Flyer, N.; Erlebacher, G., Solution to PDEs using radial basis function finite differences (RBF-FD) on multiple gpus, J. Comput. Phys., 231, 21, 7133-7151, (2012)
[15] Bayona, V.; Moscoso, M.; Kindelan, M., Optimal constant shape parameter for multiquadric based RBF-FD method, J. Comput. Phys., 230, 7384-7399, (2011) · Zbl 1343.65128
[16] Bayona, V.; Moscoso, M.; Kindelan, M., Optimal variable shape parameter for multiquadric based RBF-FD method, J. Comput. Phys., 231, 2466-2481, (2012) · Zbl 1429.65267
[17] Buhmann, M. D., Radial basis functions: theory and implementations, (2003), Cambridge University Press Cambridge · Zbl 1038.41001
[18] Bui, T. Q.; Zhang, C., Moving Kriging interpolation-based meshfree method for dynamic analysis of structures, Proc. Appl. Math. Mech., 11, 197-198, (2011)
[19] Canestrelli, A.; Dumbser, M.; Siviglia, Annunziato; Toro, E. F., Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed, Adv. Water Resour., 33, 291-303, (2010)
[20] Cao, Y.; Zhu, J.; Navon, I. M.; Luo, Z., A reduced-order approach to four-dimensional variational data assimilation using proper orthogonal decomposition, Int. J. Numer. Methods Fluids, 53, 1571-1583, (2007) · Zbl 1370.86002
[21] Chan, Y. L.; Shen, L. H.; Wu, C. T.; Young, D. L., A novel upwind-based local radial basis function differential quadrature method for convection-dominated flows, Comput. Fluids, 89, 157-166, (2014) · Zbl 1391.76529
[22] Chan, Y. L.; Shen, L. H.; Wu, C. T.; Young, D. L., Interpolation techniques for scattered data by local radial basis function differential quadrature method, Int. J. Comput. Methods, 10, (2013) · Zbl 1359.65299
[23] Chaturantabut, S., Dimension reduction for unsteady nonlinear partial differential equations via empirical interpolation methods, (2009), ProQuest
[24] Chaturantabut, S.; Sorensen, D. C., Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32, 5, 2737-2764, (2010) · Zbl 1217.65169
[25] Chaturantabut, S.; Sorensen, D. C., A state space error estimate for POD-DEIM nonlinear model reduction, SIAM J. Numer. Anal., 50, 1, 46-63, (2012) · Zbl 1237.93035
[26] Chen, W.; Fu, Z. J.; Chen, C. S., Recent advances in radial basis function collocation methods, Springer Briefs in Applied Sciences and Technology, (2014) · Zbl 1282.65160
[27] Chen, L.; Liew, K. M., A local Petrov-Galerkin approach with moving Kriging interpolation for solving transient heat conduction problems, Comput. Mech., 47, 455-467, (2011) · Zbl 1241.80005
[28] Cheng, A. H.D., Multiquadric and its shape parameter-a numerical investigation of error estimate, condition number, and round-of-error by arbitrary precision computation, Eng. Anal. Bound. Elem., 36, 220-239, (2012) · Zbl 1245.65162
[29] Cotter, C. J.; Thuburn, J., A finite element exterior calculus framework for the rotating shallow water equations, J. Comput. Phys., 257, 1506-1526, (2014) · Zbl 1351.76054
[30] Cozzolino, L.; Cimorelli, L.; Covelli, C.; Morte, R. D.; Pianese, D., The analytic solution of the shallow-water equations with partially open sluice-gates: the dam-break problem, Adv. Water Resour., 80, 90-102, (2015)
[31] Dai, B. D.; Cheng, J.; Zheng, B. J., A moving Kriging interpolation-based meshless local Petrov-Galerkin method for elastodynamic analysis, Int. J. Appl. Math. Mech., 5, 1, 1350011-1350021, (2013)
[32] Dehghan, M., Weighted finite difference techniques for the one-dimensional advection-diffusion equation, Appl. Math. Comput., 147, 2, 307-319, (2004) · Zbl 1034.65069
[33] Dehghan, M.; Abbaszadeh, M., The meshless local collocation method for solving multi-dimensional Cahn-Hilliard, Swift-Hohenberg and phase field crystal equations, Eng. Anal. Bound. Elem., 78, 49-68, (2017) · Zbl 1403.74285
[34] Dehghan, M.; Abbaszadeh, M.; Mohebbi, A., A meshless technique based on the local radial basis functions collocation method for solving parabolic-parabolic patlak-Keller-Segel chemotaxis model, Eng. Anal. Bound. Elem., 56, 1, 129-144, (2015) · Zbl 1403.65084
[35] Dehghan, M.; Salehi, R., A meshless local Petrov-Galerkin method for the time-dependent Maxwell equations, J. Comput. Appl. Math., 268, 1, 93-110, (2014) · Zbl 1293.65128
[36] Dehghan, M.; Shokri, A., A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions, Math. Comput. Simulat., 79, 3, 700-715, (2008) · Zbl 1155.65379
[37] Dehghan, M.; Mohammadi, V., The method of variably scaled radial kernels for solving two-dimensional magnetohydrodynamic (MHD) equations using two discretizations: the Crank-Nicolson scheme and the method of lines (MOL), Comput. Math. Appl., 70, 10, 2292-2315, (2015)
[38] Driscoll, T. A.; Fornberg, B., Interpolation in the limit of increasingly flat radial basis functions, Comput. Math. Appl., 43, 413-422, (2002) · Zbl 1006.65013
[39] Du, J.; Navon, I. M.; Steward, J. L.; Alekseev, A. K.; Luo, Z., Reduced-order modeling based on POD of a parabolized Navier-Stokes equation model I: forward model, Int. J. Numer. Methods Fluids, 69, 710-730, (2012)
[40] Du, J.; Navon, I. M.; Zhu, J.; Fang, F.; Alekseev, A. K., Reduced order modeling based on POD of a parabolized Navier-Stokes equations model II: trust region POD 4D VAR data assimilation, Comput. Math. Appl., 65, 380-394, (2013) · Zbl 1319.76030
[41] Duran, A.; Marche, F.; Turpault, R.; Berthon, C., Asymptotic preserving scheme for the shallow water equations with source terms on unstructured meshes, J. Comput. Phys., 287, 184-206, (2015) · Zbl 1351.76104
[42] Duran, A.; Marche, F.; Liang, Q., On the well-balanced numerical discretization of shallow water equations on unstructured meshes, J. Comput. Phys., 235, 565-586, (2013) · Zbl 1291.76215
[43] Dumbser, M.; Casulli, V., A staggered semi-implicit spectral discontinuous Galerkin scheme for the shallow water equations, Appl. Math. Comput., 219, 8057-8077, (2013) · Zbl 1366.76050
[44] Everson, R.; Sirovich, L., Karhunen-loeve procedure for gappy data, J. Opt. Soc. Am. A, 12, 8, 1657-1664, (1995)
[45] Fang, F.; Pain, C.; Navon, I.; Gorman, G.; Piggott, M.; Allison, P.; Farrell, P.; Goddard, A., A POD reduced order unstructured mesh Ocean modelling method for moderate Reynolds number flows, Ocean Model., 28, 1, 127-136, (2009)
[46] Fasshauer, G. E., Meshfree approximation methods with MATLAB, (2007), World Scientific USA · Zbl 1123.65001
[47] Felcman, J.; Kadrnka, L., Adaptive finite volume approximation of the shallow water equations, Appl. Math. Comput., 219, 3354-3366, (2012) · Zbl 1309.76059
[48] Flamant, A. A., Mecanique appliques-hydraulique, (1891), Baudry et Cie
[49] Flyer, N.; Lehto, E.; Blaise, S.; Wright, G. B.; St-Cyr, A., A guide to RBF-generated finite differences for nonlinear transport: shallow water simulations on a sphere, J. Comput. Phys., 231, 4078-4095, (2012) · Zbl 1394.76078
[50] Fornberg, B.; Lehto, E., Stabilization of RBF-generated finite difference methods for convective pdes, J. Comput. Phys., 230, 2270-2285, (2011) · Zbl 1210.65154
[51] Fornberg, B.; Lehto, E.; Powell, C., Stable calculation of Gaussian-based RBF-FD stencils, Comput. Math. Appl., 65, 627-637, (2013) · Zbl 1319.65011
[52] Gingold, R. A.; Monaghan, J. J., Smoothed particle hydrodynamics: theory and application to non-spherical stars, Mon. Not. R. Astron. Soc., 181, 375-389, (1977) · Zbl 0421.76032
[53] Gallerano, F.; Cannata, G.; Tamburrino, M., Upwind WENO scheme for shallow water equations in contravariant formulation, Comput. Fluids, 62, 1-12, (2012) · Zbl 1365.76192
[54] Giovangigli, V.; Tran, B., Mathematical analysis of a Saint-Venant model with variable temperature, Math. Models Methods Appl. Sci., 20, 8, 1251-1297, (2010) · Zbl 1204.35134
[55] Gonzalez-Rodriguez, P.; Bayona, V.; Moscoso, M.; Kindelan, M., Laurent series based RBF-FD method to avoid ill-conditioning, Eng. Anal. Bound. Elem., 52, 24-31, (2015) · Zbl 1403.65110
[56] Gu, L., Moving Kriging interpolation and element-free Galerkin method, Int. J. Numer. Methods Eng., 56, 1-11, (2003) · Zbl 1062.74652
[57] Gu, Y. T.; Wang, Q. X.; Lam, K. Y., A meshless local Kriging method for large deformation analyses, Comput. Methods Appl. Mech. Eng., 196, 1673-1684, (2007) · Zbl 1173.74471
[58] Hardy, R. L., Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res., 76, 1705-1915, (1971)
[59] Hon, Y. C.; Šarler, B.; Yun, D. F., Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface, Eng. Anal. Bound. Elem., 57, 2-8, (2015) · Zbl 1403.76140
[60] Hsu, C. T.; Yeh, K. C., Iterative explicit simulation of 1D surges and dam-break flows, Int. J. Numer. Methods Fluids, 38, 647-675, (2002) · Zbl 1098.76591
[61] Ilati, M.; Dehghan, M., Remediation of contaminated groundwater by meshless local weak forms, Comput. Math. Appl., 72, 9, 2408-2416, (2016) · Zbl 1368.76035
[62] Islam, Siraj-ul-Islam; Vertnik, R.; Šarler, B., Local radial basis function collocation method along with explicit time stepping for hyperbolic partial differential equations, Appl. Numer. Math., 67, 136-151, (2013) · Zbl 1263.65099
[63] Islam, Siraj-ul-Islam; Šarler, B.; Vertnik, R.; Kosec, G., Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled Burgers’ equations, Appl. Math. Model., 36, 1148-1160, (2012) · Zbl 1243.76076
[64] Javed, A.; Djijdeli, K.; Xing, J. T., Shape adaptive RBF-FD implicit scheme for incompressible viscous Navier-strokes equations, Comput. Fluids, 89, 38-52, (2014) · Zbl 1391.76479
[65] Jiang, T.; Zhang, Y. T., Krylov implicit integration factor WENO methods for semilinear and fully nonlinear advection-diffusion-reaction equations, J. Comput. Phys., 253, 368-388, (2013) · Zbl 1349.65305
[66] Kansa, E. J., Multiquadrics A scattered data approximation scheme with applications to computational fluid-dynamics - I, Comput. Math. Appl., 19, 127-145, (1990) · Zbl 0692.76003
[67] Kansa, E. J., Multiquadrics a scattered data approximation scheme with applications to computational fluid dynamics - II, Comput. Math. Appl., 19, 147-161, (1990) · Zbl 0850.76048
[68] Katta, K. K.; Nair, R. D.; Kumar, V., High-order finite volume shallow water model on the cubed-sphere: 1D reconstruction scheme, Appl. Math. Comput., 266, 316-327, (2015)
[69] Kerschen, G.; Golinval, J.; Vakakis, A. F.; Bergman, L. A., The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview, Nonlinear Dyn., 41, 1-3, 147-169, (2005) · Zbl 1103.70011
[70] LeVeque, R. J., Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics, (2002), Cambridge University Press · Zbl 1010.65040
[71] Li, G.; Caleffi, V.; Gao, J., High-order well-balanced central WENO scheme for pre-balanced shallow water equations, Comput. Fluids, 99, 182-189, (2014) · Zbl 1391.76484
[72] Li, X., Meshless Galerkin algorithms for boundary integral equations with moving least square approximations, Appl. Numer. Math., 61, 12, 1237-1256, (2011) · Zbl 1232.65160
[73] Li, X.; Zhu, J., A Galerkin boundary node method and its convergence analysis, J. Comput. Appl. Math., 230, 1, 314-328, (2009) · Zbl 1189.65291
[74] Lin, Z.; Xiao, D.; Fang, F.; Pain, C. C.; Navon, I. M., Non-intrusive reduced order modelling with least squares Fitting on a sparse grid, Int. J. Numer. Methods Fluids, 83, 291-306, (2017)
[75] Li, X. G.; Dai, B. D.; Wang, L. H., A moving Kriging interpolation-based boundary node method for two-dimensional potential problems, Chin. Phys. B, 19, 12, 120202-120207, (2010)
[76] Liu, G. R.; Gu, Y. T., An introduction to meshfree methods and their programming, (2005), Springer Dordrecht, Berlin, Heidelberg, New York
[77] Luo, Z. D.; Gao, J.; Xie, Z., Reduced-order finite difference extrapolation model based on proper orthogonal decomposition for two-dimensional shallow water equations including sediment concentration, J. Math. Anal. Appl., 429, 901-923, (2015) · Zbl 1318.35083
[78] Luo, Z. D.; Chen, J.; Navon, I. M.; Yang, X., Mixed finite element formulation and error estimate based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations, SIAM J. Numer. Anal., 47, 1-19, (2008)
[79] Luo, Z. D.; Chen, J.; Zhu, J.; Wang, R.; Navon, I. M., An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model, Int. J. Numer. Methods Fluids, 55, 143-161, (2007) · Zbl 1205.86007
[80] Marchenko, A. V.; Voliak, K. I., Surface wave propagation in shallow water beneath an inhomogeneous ice cover, J. Phys. Oceanogr., 27, 1602-1613, (1997)
[81] Mramor, K.; Vertnik, R.; Šarler, B., Simulation of laminar backward facing step flow under magnetic field with explicit local radial basis function collocation method, Eng. Anal. Bound. Elem., 49, 37-47, (2014) · Zbl 1403.76196
[82] Navas-Montilla, A.; Murillo, J., Energy balanced numerical schemes with very high order, the augmented roe flux ADER scheme, application to the shallow water equations, J. Comput. Phys., 290, 188-218, (2015) · Zbl 1349.76372
[83] Noelle, S.; Xing, Y.; Shu, C. W., High-order well-balanced finite volume WENO schemes for shallow water equation with moving water, J. Comput. Phys., 226, 29-58, (2007) · Zbl 1120.76046
[84] Ravindran, S., Reduced-order adaptive controllers for fluid flows using POD, J. Sci. Comput., 15, 4, 457-478, (2000) · Zbl 1048.76016
[85] Ravindran, S. S., A reduced-order approach for optimal control of fluids using proper orthogonal decomposition, Int. J. Numer. Methods Fluids, 34, 5, 425-448, (2000) · Zbl 1005.76020
[86] Rippa, S., An algorithm for selecting a good value for the parameter c in radial basis function interpolation, Adv. Comput. Math., 11, 193-210, (1999) · Zbl 0943.65017
[87] Sarra, S. A., A local radial basis function method for advection-diffusion-reaction equations on complexly shaped domains, Appl. Math. Comput., 218, 9853-9865, (2012) · Zbl 1245.65144
[88] Sarra, S. A., Adaptive radial basis function methods for time dependent partial differential equations, Appl. Numer. Math., 54, 79-94, (2005) · Zbl 1069.65109
[89] Sarra, S. A., A linear system-free Gaussian RBF method for the Gross-Pitaevskii equation on unbounded domains, Numer. Methods Partial Differ. Equ., 28, 389-401, (2012) · Zbl 1432.65136
[90] Shokri, A.; Dehghan, M., Meshless method using radial basis functions for the numerical solution of two-dimensional complex Ginzburg-Landau equation, Comput. Model. Eng. Sci., 34, 333-358, (2012) · Zbl 1357.65202
[91] Shu, C. W.; Ding, H.; Yeo, K. S., Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 192, 941-954, (2003) · Zbl 1025.76036
[92] Shu, C. W.; Ding, H.; Zhao, N., Numerical comparison of least square-based finite-difference (LSFD) and radial basis function-based finite-difference (RBF-FD) methods, Comput. Math. Appl., 51, 1297-1310, (2006) · Zbl 1146.65326
[93] Shu, C. W.; Ding, H.; Zhao, N., Numerical comparison of least square-based finite-difference (LSFD) and radial basis function-based finite-difference (RBFFD) methods, Comput. Math. Appl., 51, 8, 1297-1310, (2006) · Zbl 1146.65326
[94] Shu, C. W.; Ding, H.; Chen, H. Q.; Wang, T. G., An upwind local RBF-DQ method for simulation of inviscid compressible flows, Comput. Methods Appl. Mech. Eng., 194, 18, 2001-2017, (2005) · Zbl 1093.76052
[95] 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
[96] Ştefănescu, R.; Navon, I. M., POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations model, J. Comput. Phys., 237, 95-114, (2013) · Zbl 1286.76106
[97] Ştefănescu, R.; Sandu, A.; Navon, I. M., Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations, Int. J. Numer. Methods Fluids, 76, 8, 497-521, (2014)
[98] Tolstykh, A. I., On using RBF-based differencing formulas for unstructured and mixed structured-unstructured grid calculations, (Proceedings of the 16th IMACS, (2000), World Congress Lausanne)
[99] Tolstykh, A. I.; Shirobokov, D. A., On using radial basis functions in a finite difference mode with applications to elasticity problems, Comput. Mech., 33, 68-79, (2003) · Zbl 1063.74104
[100] Tavelli, M.; Dumbser, M., A high order semi-implicit discontinuous Galerkin method for the two-dimensional shallow water equations on staggered unstructured meshes, Appl. Math. Comput., 234, 623-644, (2014) · Zbl 1298.76120
[101] Trahan, C. J.; Dawson, C., Local time-stepping in Runge-Kutta discontinuous Galerkin finite element methods applied to the shallow-water equations, Comput. Methods Appl. Mech. Eng., 217-220, 139-152, (2012) · Zbl 1253.76065
[102] Vreugdenhil, C. B., Numerical methods for shallow-water flow, (1994), Kluwer Academic Publishers Dordrecht, Boston
[103] Wang, Y.; Navon, I. M.; Wang, X.; Cheng, Y., 2D Burgers equation with large Reynolds number using POD/DEIM and calibration, Int. J. Numer. Methods Fluids, 82, 909-931, (2016)
[104] Wazwaz, A. M., Partial differential equations and solitary waves theory, (2010), Springer Science & Business Media
[105] Wendland, H., Scattered data approximation, Cambridge monograph on applied and computational mathematics, (2005), Cambridge University Press
[106] Xing, Y., Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium, J. Comput. Phys., 257, 536-553, (2014) · Zbl 1349.76289
[107] Xing, Y.; Shu, C. W., High order finite difference WENO schemes with the exact conservation property for the shallow water equations, J. Comput. Phys., 208, 206-227, (2005) · Zbl 1114.76340
[108] Xing, Y.; Zhang, X.; Shu, C. W., Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations, Adv. Water Resour., 33, 1476-1493, (2010)
[109] Xing, Y.; Shu, C. W., High-order finite volume WENO schemes for the shallow water equations with dry states, Adv. Water Resour., 34, 1026-1038, (2011)
[110] Xiao, D.; Fang, F.; Pain, C. C.; Navon, I. M.; Salinas, P.; Muggeridge, A., Non-intrusive reduced order modeling of multi-phase flow in porous media using the POD-RBF method, J. Comput. Phys., (2015), Submitted for publication
[111] Xiao, D.; Fang, F.; Buchan, A. G.; Pain, C. C.; Navon, I. M.; Du, J.; Datum, G. H., Non-linear model reduction for the Navier-Stokes equations using residual DEIM method, J. Comput. Phys., 263, 1-18, (2014) · Zbl 1349.76288
[112] Xiao, D.; Fang, F.; Du, J.; Pain, C. C.; Navon, I. M.; Buchan, A. G.; ElSheikh, A. H.; Datum, G. H., Non-linear Petrov-Galerkin methods for reduced order modelling of the Navier-Stokes equations using a mixed finite element pair, Comput. Methods Appl. Mech. Eng., 255, 147-157, (2013) · Zbl 1297.76107
[113] Xiao, D.; Fang, F.; Pain, C. C.; Datum, G. H., Non-intrusive reduced-order modelling of the Navier-Stokes equations based on RBF interpolation, Int. J. Numer. Methods Fluids, 79, 580-595, (2015)
[114] Xiao, D.; Fang, F.; Buchan, A. G.; Pain, C. C.; Navon, I. M.; Muggeridge, A., Non-intrusive reduced order modelling of the Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 293, 522-541, (2015)
[115] Wang, Z.; Akhtar, I.; Borggaard, J.; Iliescu, T., Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison, Comput. Methods Appl. Mech. Eng., 237, 10-26, (2012) · Zbl 1253.76050
[116] Wang, Z., Reduced-order modeling of complex engineering and geophysical flows: analysis and computations, (2012), Dept of Mathematics, PhD Thesis
[117] Wang, Y.; Navon, I. M.; Wang, X.; Cheng, Y., 2D Burgers equations with large Reynolds number using POD/DEIM and calibration, Int. J. Numer. Methods Fluids, (2016)
[118] Wei, Z.; Dalrymple, R. A., SPH modeling of short-crested waves, (2017), arXiv preprint
[119] Wei, Z.; Dalrymple, R. A.; Herault, A.; Bilotta, G.; Rustico, E.; Yeh, H., SPH modeling of dynamic impact of tsunami bore on bridge piers, Coast. Eng., 104, 26-42, (2015)
[120] Wei, Z.; Dalrymple, R. A., Numerical study on mitigating tsunami force on bridges by an SPH model, J. Ocean Eng. Mar. Energy, 2, 365-380, (2016)
[121] Wei, Z.; Dalrymple, R. A.; Rustico, E.; Herault, A.; Bilotta, G., Simulation of near-Shore tsunami breaking by smoothed particle hydrodynamics method, J. Waterw. Port Coast. Ocean Eng., 142, (2016)
[122] Wei, Z.; Dalrymple, R. A., SPH modeling of vorticity generation by short-crested wave breaking, Proc. Int. Conf. - Coastal Eng. Conf., 1, (2017)
[123] Wu, W. X.; Shu, C. W.; Wang, C. M., Vibration analysis of arbitrarily shaped membranes using local radial basis function-based differential quadrature method, J. Sound Vib., 306, 252-270, (2007)
[124] Young, D. L.; Hu, S. P.; Wu, C. S., Localized radial basis function scheme for multidimensional transient generalized Newtonian fluid dynamics and heat transfer, Eng. Anal. Bound. Elem., 64, 68-89, (2016) · Zbl 1403.76143
[125] Zeng, Q. C., Silt sedimentation and relevant engineering problem-an example of natural cybernetics, (Proceedings of the Third International Congress on Industrial and Applied Mathematics, ICIAM95, (1995), Akademie Verlag Hamburg), 463-487 · Zbl 0849.76091
[126] Zhang, X.; Xiang, H., A fast meshless method based on proper orthogonal decomposition for the transient heat conduction problems, Int. J. Heat Mass Transf., 84, 729-739, (2015)
[127] Zheng, B.; Dai, B. D., A meshless local moving Kriging method for two-dimensional solids, Appl. Math. Comput., 218, 563-573, (2011) · Zbl 1275.74033
[128] Zhang, L. W.; Liew, K. M., An element-free based solution for nonlinear schrodinger equations using the ICVMLS-Ritz method, Appl. Math. Comput., 249, 333-345, (2014) · Zbl 1339.65183
[129] Zhang, L. W.; Lei, Z. X.; Liew, K. M., An element-free IMLS-Ritz framework for buckling analysis of FG-CNT reinforced composite thick plates resting on Winkler foundations, Eng. Anal. Bound. Elem., 58, 7-17, (2015) · Zbl 1403.74136
[130] Zheng, B.; Dai, B., A meshless local moving Kriging method for two-dimensional solids, Appl. Math. Comput., 218, 563-573, (2011) · Zbl 1275.74033
[131] Lucy, L., A numerical approach to the testing of fusion process, Astron. J., 82, 1013-1024, (1977)
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