A characteristic splitting mixed finite element method for three-dimensional saltwater intrusion problem
-
1723
Downloads
-
2572
Views
Authors
Jiansong Zhang
- Department of Applied Mathematics, China University of Petroleum, Qingdao 266580, China.
Jiang Zhu
- Laboratorio Nacional de Computacao Cientica, MCTI, Avenida Getulio Vargas 333, 25651-075 Petropolis, RJ, Brazil.
Danping Yang
- Department of Mathematics, East China Normal University, Shanghai 200062, China.
Hui Guo
- Department of Computational Mathematics, China University of Petroleum, Qingdao 266580, China.
Abstract
A combined method is developed for solving saltwater intrusion problem. A splitting positive definite
mixed element method is used to solve the parabolic-type water head equation and a characteristic finite
element method is used to solve the convection-diffusion type concentration equation. The convergence of
this method is considered and the optimal \(L^2\)-norm error estimate is also derived.
Share and Cite
ISRP Style
Jiansong Zhang, Jiang Zhu, Danping Yang, Hui Guo, A characteristic splitting mixed finite element method for three-dimensional saltwater intrusion problem, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 11, 5806--5820
AMA Style
Zhang Jiansong, Zhu Jiang, Yang Danping, Guo Hui, A characteristic splitting mixed finite element method for three-dimensional saltwater intrusion problem. J. Nonlinear Sci. Appl. (2016); 9(11):5806--5820
Chicago/Turabian Style
Zhang, Jiansong, Zhu, Jiang, Yang, Danping, Guo, Hui. "A characteristic splitting mixed finite element method for three-dimensional saltwater intrusion problem." Journal of Nonlinear Sciences and Applications, 9, no. 11 (2016): 5806--5820
Keywords
- Method of characteristics
- mixed finite element
- splitting system
- saltwater intrusion problem.
MSC
- 65M60
- 65M25
- 65M12
- 65M15
- 35K55
- 76M10
- 76S05
References
-
[1]
R. A. Adams, Sobolev spaces, Pure and Applied Mathematics, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975)
-
[2]
P. F. Andersen, H. O. White, J. W. Mercer, A. D. Truschel, P. S. Huyakorn, Numerical modeling of ground water flow and saltwater transport in Northern Pinellas County, Florida, In Proceedings of the Focus Conference on Southeastern Ground Water Issues. National Water Well Association, Dublin OH, 1986 (1986), 419--449
-
[3]
D. G. Ashim, N. D. Poojitha, D. Yapa, Saltwater encroachment in an aquifer, Water Res. Res., 18 (1982), 546--550
-
[4]
J. B. Bell, C. N. Dawson, G. R. Shubin, An unsplit, higher order Godunov method for scalar conservation laws in multiple dimensions, J. Comput. Phys., 74 (1988), 1--24
-
[5]
M. A. Celia, T. F. Russell, I. Herrera, R. E. Ewing, An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation, Adv. Water Res., 13 (1990), 187--206
-
[6]
P. G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York-Oxford (1978)
-
[7]
H. J. Diersch, Finite element modelling of recirculating density-driven saltwater intrusion processes in groundwater, Adv. Water Res., 11 (1988), 25--43
-
[8]
J. Douglas, Jr., T. F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19 (1982), 871--885
-
[9]
P. S. Huyakorn, P. F. Anderson, J. W. Mercer, H. O. White, Jr., Saltwater intrusion in aquifers: development and testing of a three-dimensional finite element model, Water Res. Res., 23 (1987), 293--312
-
[10]
C. Johnson, Streamline diffusion methods for problems in fluid mechanics, Finite Elem. Fluids, 6 (1986), 251--261
-
[11]
X. M. Lian, H. X. Rui, A discontinuous Galerkin method combined with mixed finite element for seawater intrusion problem, J. Syst. Sci. Complex, 23 (2010), 830--845
-
[12]
X. H. Long, Y. X. Li, Multistep characteristic finite element method for seawater intrusion problem, Numer. Math. J. Chinese Univ., 4 (2008), 325--339
-
[13]
H. X. Rui, S. C. Kim, S. D. Kim, A remark on least-squares mixed element methods for reaction-diffusion problems, J. Comput. Appl. Math., 202 (2007), 230--236
-
[14]
H. X. Rui, S. D. Kim, S. C. Kim, Split least-squares finite element methods for linear and nonlinear parabolic problems, J. Comput. Appl. Math., 223 (2009), 938--952
-
[15]
G. Segol, G. F. Pinder, W. G. Gray, A Galerkinfinite element technique for calculating the transient position of the saltwater front, Water Res. Res., 11 (1975), 343--347
-
[16]
H. Wang, An optimal-order error estimate for a family of ELLAM-MFEM approximations to porous medium flow, SIAM J. Numer. Anal., 46 (2008), 2133--2152
-
[17]
H. Wang, D. Liang, R. E. Ewing, S. L. Lyons, An ELLAM-MFEM solution technique for compressible fluid flows in porous media with point sources and sinks, J. Comput. Phys., 159 (2000), 344--376
-
[18]
H. Wang, D. Liang, R. E. Ewing, S. L. Lyons, G. Qin, An ELLAM approximation for highly compressible multicomponent flows in porous media Locally conservative numerical methods for flow in porous media, Comput. Geosci., 6 (2002), 227--251
-
[19]
Y.-Q. Xue, C.-H. Xie, J. C. Wang, Study on joint-surface of saltwater and freshwater in seawater intrusion problem, Nanjing University Press, Nanjing (1991)
-
[20]
D.-P. Yang, Analysis of least-squares mixed finite element methods for nonlinear nonstationary convection- diffusion problems, Math. Comp., 69 (2000), 929--963
-
[21]
D.-P. Yang, A splitting positive definite mixed element method for miscible displacement of compressible flow in porous media, Numer. Methods Partial Differential Equations, 17 (2001), 229--249
-
[22]
Y. R. Yuan, Characteristic finite element methods for positive semidefinite problem of two phase miscible flow in three dimensions, Chinese Sci. Bull., 22 (1996), 2027--2032
-
[23]
Y. R. Yuan, N. Du, Y.-J. Han, Careful numerical simulation and analysis of migration-accumulation of Tanhai Region, Appl. Math. Mech., 26 (2005), 741--752
-
[24]
Y. R. Yuan, D. Liang, H. X. Rui, Characteristics-finite element methods for seawater intrusion numerical simulation and theoretical analysis, Acta Math. Appl. Sin., 14 (1998), 11--23
-
[25]
Y. R. Yuan, D. Liang, H. X. Rui, Numerical method and simulation of seawater intrusion and protection projects, Chinese J. Comput. Phys., 18 (2001), 556--562
-
[26]
Y. R. Yuan, D. Liang, H. X. Rui, Predicting the consequences of seawater intrusion and protection projects, Appl. Math. Mech., 22 (2001), 1291--1300
-
[27]
Y. R. Yuan, D. Liang H. X. Rui, The modified method of upwind with finite difference fractional steps procedure for the numerical simulation and analysis of seawater intrusion, Progr. Natur. Sci. (English Ed.), 16 (2006), 1127--1140
-
[28]
Y. R. Yuan, D. Liang, H. X. Rui, G. H. Wang, The characteristics finite difference method for seawater intrusion numerical simulation and optimal order \(L^2\) error estimates, Acta Math. Appl. Sin., 19 (1996), 395--404
-
[29]
Y. R. Yuan, H. X. Rui, D. Liang, C. F. Li, The theory and application of upwind finite difference fractional steps procedure for seawater intrusion, Int. J. Geosci., 3 (2012), 972--991
-
[30]
Z.-Y. Zhang, The alternating-direction schemes and numerical analysis for the three-dimensional seawater intrusion simulation, Acta Math. Appl. Sin. Engl. Ser., 18 (2002), 389--404
-
[31]
J. S. Zhang, H. Guo, A split least-squares characteristic mixed element method for nonlinear nonstationary convection-diffusion problem, Int. J. Comput. Math., 89 (2012), 932--943
-
[32]
J. S. Zhang, D.-P. Yang, A fully-discrete splitting positive definite mixed finite element scheme for compressible miscible displacement in porous media, J. Shandong Univ. Nat. Sci., 41 (2006), 1--10
-
[33]
J. S. Zhang, D.-P. Yang, A splitting positive definite mixed element method for second-order hyperbolic equations, Numer. Methods Partial Differential Equations, 25 (2009), 622--636
-
[34]
J. S. Zhang, D.-P. Yang, S. Q. Shen, J. Zhu, A new MMOCAA-MFE method for compressible miscible displacement in porous media, Appl. Numer. Math., 80 (2014), 65--80
-
[35]
J. Zhu, The characteristic numerical methods for the KdV equation, I, (Chinese) Numer. Math. J. Chinese Univ., 10 (1988), 11--27
-
[36]
J. Zhu, The characteristic numerical methods for nonlinear RLW equations, Acta Math. Appl. Sin., 13 (1990), 64--73