Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-04T21:50:17.916Z Has data issue: false hasContentIssue false
  • English
  • Français

A Combined Discontinuous Galerkin Method for Saltwater Intrusion Problem with Splitting Mixed Procedure

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  17 January 2017

Jiansong Zhang ,
Jiang Zhu  and
Danping Yang
Jiansong Zhang*
Affiliation:
Department of Applied Mathematics, China University of Petroleum, 66 Changjiang West Road, Qingdao 266580, China
Jiang Zhu*
Affiliation:
Laboratório Nacional de Computação Científica, MCTI, Avenida Getúlio Vargas 333, 25651-075 Petrópolis, RJ, Brazil
Danping Yang*
Affiliation:
Department of Mathematics, East China Normal University, Shanghai 200062, China
*
*Corresponding author. Email:[email protected] (J. Zhang), [email protected] (J. Zhu), [email protected] (D. Yang)
*Corresponding author. Email:[email protected] (J. Zhang), [email protected] (J. Zhu), [email protected] (D. Yang)
*Corresponding author. Email:[email protected] (J. Zhang), [email protected] (J. Zhu), [email protected] (D. Yang)
Get access

Abstract

In this paper, a new combined method is presented to simulate saltwater intrusion problem. A splitting positive definite mixed element method is used to solve the water head equation, and a symmetric discontinuous Galerkin (DG) finite element method is used to solve the concentration equation. The introduction of these two numerical methods not only makes the coefficient matrixes symmetric positive definite, but also does well with the discontinuous problem. The convergence of this method is considered and the optimal L2-norm error estimate is also derived.

Keywords

Splitting mixed systemdiscontinuous Galerkin methodsaltwater intrusion problemconvergence analysis

MSC classification

Secondary: 65M12: Stability and convergence of numerical methods 65M15: Error bounds 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 3 , June 2017 , pp. 651 - 666
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Adams, R. A., Sobolev Spaces, Academic, New York, 1975.Google Scholar
[2] Ackerer, P., Younes, A. and Mose, R., Modeling variable density flow and solute transport in porous medium: I. Numerical model and verification, Transport Porous Media, 35(3) (1999), pp. 345–373.Google Scholar
[3] Arnold, D. N., An interior penalty finite element method with discontinuous element, SIAM J. Numer. Anal., 19(4) (1982), pp. 742–760.Google Scholar
[4] Bell, J. B., Dawson, C. N. and Shubin, G. R., An unsplit high-order Godunov scheme for scalar conservation laws in two dimensions, J. Comput. Phys., 74 (1988), pp. 1–24.Google Scholar
[5] Bevilacqua, L., Feijoo, R. and Rojas M., L. F., A variational principle for the Laplace's operator with application in the torsion of composite rods, Int. J. Solids Strucrures, 10 (1974), pp. 1091–1102.CrossRefGoogle Scholar
[6] Bues, M. A. and Oltean, C., Numerical simulations for saltwater intrusion by the mixed hybrid finite element method and discontinuous finite element method, Transport Porous Media, 40(2) (2000), pp. 171–200.Google Scholar
[7] Ciarlet, P. G., The Finite Element Methods for Elliptic Problems, North-Holland, New York, 1978.CrossRefGoogle Scholar
[8] Cooper, H. H. Jr., A hypothesis concerning the dynamic balance of fresh water and salt water in a coastal aquifer, Sea water in coastal aquifer, Geological survey water-supply paper 1613-C, US Geological Survey, 1964, pp. 1–12.Google Scholar
[9] Cui, M. R., Analysis of a semidiscrete discontinuous Galerkin scheme for compressible miscible displacement problem, J. Comput. Appl. Math., 214 (2008), pp. 617–636.Google Scholar
[10] Fraeijs De Veubeke, B., Displacement and equilibrium models in the finite element method, In Zienkiewicz, O. C. and Holister, G., editors, Stress Analysis, JohnWiley and Sons, New York, 1965.Google Scholar
[11] Frind, E. O., Simulation of long-term transient density-dependent transport in groundwater, Adv. Water Resour., 5(2) (1982), pp. 73–88.CrossRefGoogle Scholar
[12] Huyakorn, P. S. and Pinder, G. F., Computational Methods in Subsurface Flow, Academic Press, 1983.Google Scholar
[13] Johnson, C., Streamline Diffusion Methods for Problems in Fluid Mechanics, in: Finite Element in Fluids VI, Wiley, New York, 1986.Google Scholar
[14] Kohout, R. A., The flow of fresh water and salt water in the Biscayne aquifer of the Miami area, Florida, Sea water in coastal aquifer, Geological survey water supply paper 1613-C, US Geological Survey, 1964, pp. 12–32.Google Scholar
[15] Lian, X. M. and Rui, H. X., A discontinuous Galerkin method combined with mixed finite element for seawater intrusion problem, J. Syst. Sci. Complex, 23 (2010), pp. 830–845.Google Scholar
[16] Li, X. G., Zhu, J., Zhang, R. P. and Cao, S., A combined discontinuous Galerkin method for the dipolar Bose-Einstein condensation, J. Comput. Phys., 275 (2014), pp. 363–376.Google Scholar
[17] Li, Z., Yu, X., Zhu, J. and Jia, Z., A Runge-Kutta discontinuous Galerkin method for Lagrangian compressible Euler equations in two-dimensions, Commun. Comput. Phys., 15 (2014), pp. 1184–1206.Google Scholar
[18] Long, X. H. and Li, Y. X., Multistep characteristic finite element method for seawater intrusion problem, Numer. Math. J. China University, 30(4) (2008), pp. 325–339.Google Scholar
[19] Pinder, G. F. and Cooper, H. A., Numerical technique for calculating the transient position of the saltwater front, Water Resour. Res., 6 (1970), pp. 875–882.CrossRefGoogle Scholar
[20] Reed, W. H. and Hill, T. R., Triangular mesh methods for the neutron transport equation, Tech. Report No. LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, 1973.Google Scholar
[21] Rivire, B. and Wheeler, M. F., Discontinuous Galerkin methods for flow and transport problem in porous media, Commun. Numer. Methods Eng., 18 (2002), pp. 63–68.Google Scholar
[22] Segol, G., Pinder, G. F. and Gray, W. G., A Galerkin finite element technique for calculating the transient position of the saltwater front, Water Resour. Res., 11(2) (1975), pp. 343–347.Google Scholar
[23] Segol, G., Classic Groundwater Simulations: Proving and Improving Numerical Models, New Jersey, Prentice Hall, 1994.Google Scholar
[24] Sun, S. Y., Riviera, B. and Wheeler, M. F., A combined mixed finite element and discontinuous Galerkin method for miscible displacement problem in porous media, Recent Progress in Computational and Applied PDES, (2002), pp. 323–351.Google Scholar
[25] Sun, S. Y. and Wheeler, M. F., Discontinuous Galerkin methods for coupled flow and reactive transport problems, Appl. Numer. Math., 52 (2005), pp. 273–298.Google Scholar
[26] Voss, C. I. and Souza, W. R., Variable density flow and solute transport simulation of regional aquifers containing a narrow freshwater-saltwater transition zone, Water Resour. Res., 23(10) (1987), pp. 1851–1866.Google Scholar
[27] Yang, D. P., Analysis of least-squares mixed finite element methods for nonlinear nonstationary convection-diffusion problems, Math. Comput., 69 (2000), pp. 929–963.Google Scholar
[28] Yang, D. P., A splitting positive definite mixed element method for miscible displacement of compressible flow in porous media, Numer. Meth. Part. Differ. Eq., 17 (2001), pp. 229–249.Google Scholar
[29] Yuan, Y. R., Du, N., Han, Y. J., Careful numerical simulation and analysis of migration-accumulation of Tanhai Region, Appl. Math. Mech., 26(6) (2005), pp. 741–752.Google Scholar
[30] Yuan, Y. R., Liang, D., Rui, H. X. and Wang, G. H., The characteristics-finite difference methods of sea water intrusion numerical simulation and optimal order l2 error estimates, Acta Math. Appl. Sinica, 19(3) (1996), pp. 395–404.Google Scholar
[31] Yuan, Y. R., Characteristic finite element methods for positive semidefinite problem of two phase miscible flow in three dimensions, Sci. Bull China, 22 (1996), pp. 2027–2032.Google Scholar
[32] Zhang, J. S. and Guo, H., A split least-squares characteristic mixed element method for nonlinear nonstationary convection-diffusion problem, Inter. J. Comput. Math., 89(7) (2012), pp. 932–943.Google Scholar
[33] Zhang, J. S., Yang, D. P., Shen, S. Q. and Zhu, J., A new MMOCAA-MFE method for compressible miscible displacement in porous media, Appl. Numer. Math., 80 (2014), pp. 65–80.CrossRefGoogle Scholar
[34] Zhang, J. S. and Yang, D. P., A fully-discrete splitting positive definite mixed finite element scheme for compressible miscible displacement in porous media, J. Shandong University (Nature Science), 41 (2006), pp. 1–10.Google Scholar
[35] Zhang, J. S. and Yang, D. P., A splitting positive definite mixed element method for second order hyperbolic equations, Numer. Meth. Part. Differ. Eq., 25(3) (2009), pp. 622–636.CrossRefGoogle Scholar
[36] Zhang, R. P., Yu, X., Zhu, J. and Loula, A. F. D., Direct discontinuous Galerkin method for nonlinear reaction-diffusion systems in pattern formation, Appl. Math. Model., 38 (2014), pp. 1612–1621.Google Scholar
[37] Zhu, J., Yu, X. and Loula, A. F. D., Mixed discontinuous Galerkin analysis of thermally nonlinear coupled problem, Comput. Methods Appl. Mech. Eng., 200 (2011), pp. 1479–1489.Google Scholar
[38] Zhu, J., The characteristic numerical methods for KdV equation, Numer. Math. J. China University, 10 (1988), pp. 11–27.Google Scholar
[39] Zhu, J., The characteristic numerical methods for RLW equation, Acta Math. Appl. Sinica, 13 (1990), pp. 64–73.Google Scholar