A mass-conservative characteristic splitting mixed element method for saltwater intrusion problem

Volume 10, Issue 11, pp 6102--6118 http://dx.doi.org/10.22436/jnsa.010.11.44

Publication Date: November 30, 2017 Submission Date: December 27, 2016

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1785 Downloads
• 2966 Views

Authors

Jiansong Zhang - Department of Applied Mathematics, China University of Petroleum, Qingdao 266580, China. Yuezhi Zhang - Department of Applied Mathematics, China University of Petroleum, Qingdao 266580, China. Zhaohui Liu - Department of Applied Mathematics, China University of Petroleum, Qingdao 266580, China.

Abstract

A new characteristic mixed finite element method is developed for solving saltwater intrusion problem. In this algorithm, the splitting mixed finite element (SMFE) method is applied for solving the parabolic-type water head equation, and the mass-conservative characteristic (MCC) finite element method is applied for solving the convection-diffusion type concentration equation. The application of the splitting mixed element method results in a symmetric positive definite coefficient matrix of the mixed element system and separating the flux equation from the water head equation. While the mass-conservative characteristic finite element method does well in handling convection-dominant diffusion problem and keeps mass balance. The convergence of this method is considered and the optimal \(L^2\)-norm error estimate is also derived.

Share and Cite

Keywords

• Method of characteristics
• mass-conservative
• splitting mixed finite element
• error estimate
• saltwater intrusion problem

MSC

•  65M12
•  65M15
•  65M25
•  65M60

References

• [1] R. A. Adams, Sobolev spaces, Pure and Applied Mathematics, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975)
  • Google Scholar


• [2] P. F. Anderson Jr., H. O. White, J. W. Mercer, P. S. Huyakorn, Numerical modeling of groundwater flow and saltwater transport in northern pinellas country, Southwest Florida Water Management Distric, Florida (1985)
  • Google Scholar


• [3] 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.
  • View Article
  • Google Scholar
  • MATH


• [4] M. A. Cella, T. F. Russell, I. Herrera, R. E. Ewing, An Eulerian-Lagrangian localized adjoint method for the advectiondiffusion equation, Adv. Water Resour., 13 (1990), 187–206.
  • View Article
  • Google Scholar


• [5] 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)
  • Google Scholar


• [6] H. J. Diersch, Finite element modelling of recirculating density-driven saltwater intrusion processes in groundwater , Adv. Water Resour., 11 (1988), 25–43.
  • View Article
  • Google Scholar


• [7] 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.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] A. D. Gupta, P. N. Yapa , Saltwater encroachment in an aquifer: A case study, Water Resour. Res., 18 (1982), 546–556.
  • View Article
  • Google Scholar


• [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 Resour. Res., 23 (1987), 293–312.
  • View Article
  • Google Scholar


• [10] C. Johnson , Streamline diffusion methods for problems in fluid mechanics, Finite Elem. Fluids, 6 (1986), 251–261.
  • Google Scholar


• [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.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] H.-X. Rui, M. Tabata, A mass-conservative characteristic finite element scheme for convection-diffusion problems, J. Sci. Comput., 43 (2010), 416–432.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] G. Segol, G. F. Pinder, W. G. Gray , A Galerkin finite element technique for calculating the transient position of the saltwater front, Water Resour. Res., 11 (1975), 343–347.
  • View Article
  • Google Scholar


• [14] 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.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] 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), 43–94.
  • Google Scholar


• [16] D.-P. Yang, Analysis of least-squares mixed finite element methods for nonlinear nonstationary convection-diffusion problems, Math. Comp., 69 (2000), 929–963.
  • View Article
  • MathSciNet
  • Google Scholar


• [17] 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.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] 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.
  • View Article
  • Google Scholar
  • MATH


• [19] Y.-R. Yuan, D. Liang, H.-X. Rui , Characteristics-finite element methods for seawater intrusion numerical simulation and theoretical analysis, Acta Math. Appl. Sinica (English Ser.), 14 (1998), 11–23.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] Y.-R. Yuan, D. Liang, H.-X. Rui , Numerical method and simulation of seawater intrusion and protection projects, Chinese J. Comput. Phys., 18 (2001), 562–567.
  • Google Scholar


• [21] Y.-R. Yuan, D. Liang, H.-X. Rui, Predicting the consequences of seawater intrusion and protection projects, Appl. Math. Mech., 22 (2001), 1291–1300.
  • View Article
  • Google Scholar
  • MATH


• [22] Y.-R. Yuan, D. Liang, H.-X. Rui, G.-H. Wang , Characteristic-difference methods for the numerical simulation of sea water intrusion and optimal order \(l^2\) error estimates, (Chinese) Acta Math. Appl. Sinica, 19 (1996), 395–404.
  • MathSciNet


• [23] 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.
  • View Article
  • Google Scholar


• [24] 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.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] 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.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] 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.
  • Google Scholar


• [27] 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.
  • View Article
  • MathSciNet
  • Google Scholar


• [28] 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.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] Y.-Z. Zhang, J.-S. Zhang, The splitting mixed element method for parabolic equation and its application in chemotaxis model , Appl. Math. Comput., 313 (2017), 287–300.
  • View Article
  • MathSciNet
  • Google Scholar


• [30] J.-S. Zhang, J. Zhu, R.-P. Zhang, Characteristic splitting mixed finite element analysis of Keller-Segel chemotaxis models, Appl. Math. Comput., 278 (2016), 33–44.
  • View Article
  • MathSciNet
  • Google Scholar