Analysis and application of the interpolating element free Galerkin (IEFG) method to simulate the prevention of groundwater contamination with application in fluid flow. (English) Zbl 1433.65233
Summary: We develop a meshless numerical procedure to simulate the groundwater equation (GWE). The used technique is based on the interpolating element free Galerkin (IEFG) method. The interpolating moving least squares (IMLS) approximation produces a set of functions such that they are well-known as “shape functions”. The IEFG technique employs the shape functions of IMLS approximation. The shape functions of IMLS approximation vanish on the boundary and also they satisfy the property of the Kronecker Delta function. Thus, Dirichlet boundary conditions can be exactly imposed. In this paper, we check the unconditional stability and convergence of the proposed numerical scheme based on the energy method. The numerical results confirm the theoretical analysis.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65D32 | Numerical quadrature and cubature formulas |
| 86A05 | Hydrology, hydrography, oceanography |
| 86-10 | Mathematical modeling or simulation for problems pertaining to geophysics |
| 76M99 | Basic methods in fluid mechanics |
| 76T30 | Three or more component flows |
Keywords:
groundwater equation and fluid flow; prevention of groundwater contamination; element free Galerkin (EFG) method; interpolating MLS; unconditionally stable; convergenceReferences:
| [1] | Belytschko, T.; Lu, Y. Y.; Gu, L., Element free Galerkin methods, Internat. J. Numer. Methods Engrg., 37, 229-256 (1994) · Zbl 0796.73077 |
| [2] | Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: An overview and recent developments, Comput. Methods Appl. Mech. Engrg., 139, 3-47 (1996) · Zbl 0891.73075 |
| [3] | Chung, H. J.; Belytschko, T., An error estimate in the EFG method, Comput. Mech., 21, 91-100 (1998) · Zbl 0910.73060 |
| [4] | Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares methods, Math. Comp., 37, 155, 141-158 (1981) · Zbl 0469.41005 |
| [5] | Ren, H. P.; Zhang, W., An improved boundary element-free method (IBEFM) for two-dimensional potential problems, Chin. Phys. B, 18, 10, 4065-4073 (2009) |
| [6] | Ren, H. P.; Cheng, Y. M.; Zhang, W., An interpolating boundary element-free method (IBEFM) for elasticity problems, Sci. China Phys. Mech. Astron., 53, 4, 758-766 (2010) |
| [7] | Sun, F.; Wang, J.; Cheng, Y. M., An improved interpolating element-free Galerkin method for elasticity, Chin. Phys. B, 22, 12, 120-203 (2013) |
| [8] | Wang, J.; Wang, J.; Sun, F.; Cheng, Y. M., An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems, Int. J. Comput. Methods, 10, 1350043 (2013) · Zbl 1359.65282 |
| [9] | Wang, J.; Sun, F.; Cheng, Y. M.; Huang, A., Error estimates for the interpolating moving least-squares method, Appl. Math. Comput., 245, 321-342 (2014) · Zbl 1335.65018 |
| [10] | Sun, F.; Wang, J.; Cheng, Y. M.; Huang, A., Error estimates for the interpolating moving least-squares method in n-dimensional space, Appl. Numer. Math., 98, 79-105 (2015) · Zbl 1329.65280 |
| [11] | Ren, H. P.; Cheng, J.; Huang, A., The complex variable interpolating moving least-squares method, Appl. Math. Comput., 219, 1724-1736 (2012) · Zbl 1291.65138 |
| [12] | Cheng, Y. M.; Bai, F.; Peng, M., A novel interpolating element-free Galerkin (IEFG) method for two-dimensional elastoplasticity, Appl. Math. Model., 38, 21, 5187-5197 (2014) · Zbl 1449.74196 |
| [13] | Deng, Y.; Liu, C.; Peng, M.; Cheng, Y. M., The interpolating complex variable element-free Galerkin method for temperature field problems, Int. J. Appl. Mech., 7, 1550017 (2015) |
| [14] | Cheng, Y. M.; Bai, F.; Liu, C.; Peng, M., Analyzing nonlinear large deformation with an improved element-free Galerkin method via the interpolating moving least-squares method, Int. J. Comput. Mater. Sci. Eng., 5, 1650023 (2016) |
| [15] | Liew, K. M.; Cheng, Y. M., Complex variable boundary element-free method for two-dimensional elastodynamic problems, Comput. Methods Appl. Mech. Eng., 198, 3925-3933 (2009) · Zbl 1231.74502 |
| [16] | Chen, L.; Cheng, Y. M., The complex variable reproducing kernel particle method for bending problems of thin plates on elastic foundations, Comput. Mech., 62, 67-80 (2018) · Zbl 1461.74089 |
| [17] | Meng, Z. J.; Cheng, H.; Ma, L. D.; Cheng, Y. M., The dimension split element-free Galerkin method for three-dimensional potential problems, Acta Mech. Sin., 34, 3, 462-474 (2018) · Zbl 1404.65275 |
| [18] | Dehghan, M.; Abbaszadeh, M., Variational multiscale element-free Galerkin method combined with the moving kriging interpolation for solving some partial differential equations with discontinuous solutions, Comput. Appl. Math., 37, 3, 3869-3905 (2018) · Zbl 1404.65084 |
| [19] | Liu, F. B.; Cheng, Y. M., The improved element-free Galerkin method based on the nonsingular weight functions for elastic large deformation problems, Int. J. Comput. Mater. Sci. Eng., 7, 3, 1850023 (2018) |
| [20] | Liu, F. B.; Cheng, Y. M., The improved element-free Galerkin method based on the nonsingular weight functions for inhomogeneous swelling of polymer gels, Int. J. Appl. Mech., 10, 4, 1850047 (2018) |
| [21] | Liu, F. B.; Wu, Q.; Cheng, Y. M., A meshless method based on the nonsingular weight functions for elastoplastic large deformation problems, Int. J. Appl. Mech., 11, 1, 1950006 (2019) |
| [22] | Feng-Xin, S.; Ju-Feng, W.; Yu-Min, C., An improved interpolating element-free Galerkin method for elasticity, Chin. Phys. B, 22, 12, Article 120203 pp. (2013) |
| [23] | Ju-Feng, W.; Feng-Xin, S.; Yu-Min, C., An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems, Chin. Phys. B, 21, 9, Article 090204 pp. (2012) |
| [24] | Ren, H.; Cheng, Y., The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems, Eng. Anal. Bound. Elem., 36, 5, 873-880 (2012) · Zbl 1352.65539 |
| [25] | Li, D.; Bai, F.; Cheng, Y.; Liew, K. M., A novel complex variable element-free Galerkin method for two-dimensional large deformation problems, Comput. Methods Appl. Mech. Eng., 233, 1-10 (2012) · Zbl 1253.74106 |
| [26] | Zhao, N.; Ren, H., The interpolating element-free Galerkin method for 2D transient heat conduction problems, Math. Probl. Eng., 2014 (2014), Article ID 712834, 9 pages · Zbl 1407.80009 |
| [27] | Dehghan, M.; Abbaszadeh, M., Proper orthogonal decomposition variational multiscale element free Galerkin (POD-VMEFG) meshless method for solving incompressible Navier-Stokes equation, Comput. Methods Appl. Mech. Engrg., 311, 856-888 (2016) · Zbl 1439.76060 |
| [28] | Li, X., A meshless interpolating Galerkin boundary node method for Stokes flows, Eng. Anal. Bound. Elem., 51, 112-122 (2015) · Zbl 1403.76090 |
| [29] | Zhang, L.; Deng, Y.; Liew, K. M.; Cheng, Y., The improved complex variable element-free Galerkin method for two-dimensional Schrodinger equation, Comput. Math. Appl., 68, 10, 1093-1106 (2014) · Zbl 1367.35141 |
| [30] | Zhang, L.; Deng, Y.; Liew, K. M., An improved element-free Galerkin method for numerical modeling of the biological population problems, Eng. Anal. Bound. Elem., 40, 181-188 (2014) · Zbl 1297.65123 |
| [31] | Li, D.; Zhang, Z.; Liew, K. M., A numerical framework for two-dimensional large deformation of inhomogeneous swelling of gels using the improved complex variable element-free Galerkin method, Comput. Methods Appl. Mech. Eng., 274, 84-102 (2014) · Zbl 1296.74120 |
| [32] | Li, X.; Wang, Q., Analysis of the inherent instability of the interpolating moving least squares method when using improper polynomial bases, Eng. Anal. Bound. Elem., 73, 21-34 (2016) · Zbl 1403.65206 |
| [33] | Abbaszadeh, M.; Dehghan, M., Numerical and analytical investigations for neutral delay fractional damped diffusion-wave equation based on the stabilized interpolating element free Galerkin (IEFG) method, Appl. Numer. Math., 145, 488-506 (2019) · Zbl 1428.65073 |
| [34] | Abbaszadeh, M.; Dehghan, M., The interpolating element-free Galerkin method for solving Korteweg de Vries-Rosenau-regularized long-wave equation with error analysis, Nonlinear Dynam., 96, 1345-1365 (2019) · Zbl 1437.65121 |
| [35] | Dehghan, M.; Abbaszadeh, M., Error analysis and numerical simulation of magnetohydrodynamics (MHD) equation based on the interpolating element free Galerkin (IEFG) method, Appl. Numer. Math., 137, 252-273 (2019) · Zbl 1412.65070 |
| [36] | Dehghan, M.; Abbaszadeh, M.; Mohebbi, A., Analysis of two methods based on Galerkin weak form for fractional diffusion-wave: Meshless interpolating element free Galerkin (IEFG) and finite element methods, Eng. Anal. Bound. Elem., 64, 205-221 (2016) · Zbl 1403.65068 |
| [37] | Abbaszadeh, M.; Dehghan, M., The two-grid interpolating element free Galerkin (TG-IEFG) method for solving Rosenau-regularized long wave (RRLW) equation with error analysis, Appl. Anal., 97, 1129-1153 (2018) · Zbl 1395.65133 |
| [38] | Dehghan, M.; Abbaszadeh, M., Interpolating stabilized moving least squares (MLS) approximation for 2D elliptic interface problems, Comput. Methods Appl. Mech. Engrg., 328, 775-803 (2018) · Zbl 1439.82015 |
| [39] | Dehghan, M.; Abbaszadeh, M., An upwind local radial basis functions-differential quadrature (RBF-DQ) method with proper orthogonal decomposition (POD) approach for solving compressible Euler equation, Eng. Anal. Bound. Elem., 92, 244-256 (2018) · Zbl 1403.65109 |
| [40] | Mirzaei, D.; Schaback, R.; Dehghan, M., On generalized moving least squares and diffuse derivatives, IMA J. Numer. Anal., 32, 983-1000 (2012) · Zbl 1252.65037 |
| [41] | Salehi, R.; Dehghan, M., A generalized moving least square reproducing kernel method, J. Comput. Appl. Math., 249, 120-132 (2013) · Zbl 1285.65080 |
| [42] | Salehi, R.; Dehghan, M., A moving least square reproducing polynomial meshless method, Appl. Numer. Math., 69, 34-58 (2013) · Zbl 1284.65137 |
| [43] | Hon, Y.-C.; Šarler, B.; D.-f. Yun, M., 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 |
| [44] | Li, Q.; Chen, S.; Luo, X., Steady heat conduction analyses using an interpolating element-free Galerkin scaled boundary method, Appl. Math. Comput., 300, 103-115 (2017) · Zbl 1411.80004 |
| [45] | 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 |
| [46] | H.E. Kobus, W. Kinzelbach, Contaminant Transport in Groundwater: Proceedings of an international symposium, Stuttgart, 4-6 1989, vol. 3, CRC Press, 1989. |
| [47] | Liu, W.; Huang, J.; Long, X., Coupled nonlinear advection-diffusion-reaction system for prevention of groundwater contamination by modified upwind finite volume element method, Comput. Math. Appl., 69, 6, 477-493 (2015) · Zbl 1444.65056 |
| [48] | Tambue, A., An exponential integrator for finite volume discretization of a reaction-advection-diffusion equation, Comput. Math. Appl., 71, 9, 1875-1897 (2016) · Zbl 1443.65167 |
| [49] | Wazwaz, A.-M., Partial Differential Equations and Solitary Waves Theory (2010), Springer Science & Business Media |
| [50] | Ilati, M.; Dehghan, M., Remediation of contaminated groundwater by meshless local weak forms, Comput. Math. Appl., 72, 9, 2408-2416 (2016) · Zbl 1368.76035 |
| [51] | Sarra, S. A., A local radial basis function method for advection-diffusion-reaction equations on complexly shaped domains, Appl. Math. Comput., 218, 19, 9853-9865 (2012) · Zbl 1245.65144 |
| [52] | Budinski, L.; Fabian, J.; Stipić, M., Lattice Boltzmann method for groundwater flow in non-orthogonal structured lattices, Comput. Math. Appl., 70, 10, 2601-2615 (2015) · Zbl 1443.76166 |
| [53] | D’Acunto, B.; Parente, F.; Urciuoli, G., Numerical models for 2d free boundary analysis of groundwater in slopes stabilized by drain trenches, Comput. Math. Appl., 53, 10, 1615-1626 (2007) · Zbl 1152.65460 |
| [54] | 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 |
| [55] | Liu, B., An error analysis of a finite element method for a system of nonlinear advection-diffusion-reaction equations, Appl. Numer. Math., 59, 8, 1947-1959 (2009) · Zbl 1169.65085 |
| [56] | Wang, X.; Huo, Z.; Feng, S.; Guo, P.; Guan, H., Estimating groundwater evapotranspiration from irrigated cropland incorporating root zone soil texture and moisture dynamics, J. Hydrol., 543, 501-509 (2016) |
| [57] | Zhang, W.; Yang, H.; Fang, L.; Cui, P.; Fang, Z., Study on heat transfer of pile foundation ground heat exchanger with three-dimensional groundwater seepage, Int. J. Heat Mass Transfer, 105, 58-66 (2017) |
| [58] | Shoushtari, S. M.H. J.; Cartwright, N.; Perrochet, P.; Nielsen, P., Two-dimensional vertical moisture-pressure dynamics above groundwater waves: Sand flume experiments and modelling, J. Hydrol., 544, 467-478 (2017) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
