Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A spacetime discontinuous Galerkin method for the two-dimensional unsteady convection-dispersion problems

  • 105 Accesses

Abstract

The unsteady two-dimensional convection-diffusion (CD) equation, which is the governing equation of the unsteady two-dimensional convection-dispersion problem, as the water contamination problems, has a mixed hyperbolic-parabolic character. When the equation has a strong mixed hyperbolic character, the exact solution is nonsmooth. In this case, the conventional numerical methods give approximate solutions which either oscillate or smear out the sharp front of the exact solution. The spacetime discontinuous Galerkin method (STDGM) is an extention of the space discontinuous Galerkin method (SDGM), applying the discontinuity in the time direction, as well as in space. Both these methods are respective modifications of the standard Galerkin finite element method (SGM). In this paper, the STDGM is applied to solve the CD equation, when the Péclet number has extremely high values, which means a strong mixed hyperbolic character. With this method, three artificial diffusion terms are introduced by modifying the test functions of the finite element method. These functions include the discontinuity int, x andy axis. The results obtained from the analytical solution of the problem are used for testing the numerical solution, applying both the space-discontinuous Galerkin method (SDGM) and the STDGM and are presented in diagrams, from which useful observations, comparisons and conclusions can be drawn.

This is a preview of subscription content, log in to check access.

References

  1. Antonopoulos, V. and Papazafiriou, Z., 1990, Solutions of one-dimensional water flow and mass transport equations in variably saturated porous media by the finite element method,J. Hydrology 119, 151–167.

  2. Asadzadeh, M., 1986, Convergence analysis of some numerical methods for neutron transport and vlasov equations, PhD thesis, Chalmers Univ. of Technology, Göteborg.

  3. Babuška, I., 1971, Error-bounds for finite element method,Numer. Math. 16, 322–333.

  4. Babuška, I., Zienkiewicz, O. C., Gago, J., and Oliveira, E. R. de A. (Eds.), 1986,Accuracy Estimates and Adaptive Refinements in Finite Element Computations, Wiley, New York.

  5. Bramble, J. H. and Hilbert, S. R., 1970, Estimation of linear functional on Sobolev spaces with application to Fourier transforms and spline interpolation,SIAM J. Numer. Anal. 7(1), 112–124.

  6. Brooks, A. N. and Hughes, T. J. R., 1982, Streamline upwind / Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations,Comp. Methods Appl. Mech. Engng. 30, (CMA 760), 1–61.

  7. Cohen, M. F., 1983, Application of the Petrov-Galerkin method to chemical-flooding reservoir simulation in one dimension,Comp. Methods Appl. Mech. Engng. 41, 195–218.

  8. Huyakorn, P. S. and Pinder, G. F., 1983,Computational Methods in Subsurface Flow, Academic Press, New York.

  9. Johnson, C., Nävert, U. and Pitkäranta, J., 1985, Finite element methods for linear hyperbolic problems,Comp. Methods Appl. Mech. Engng. 45, 285–312.

  10. Johnson, C. and Pitkäranta, J., 1983, Convergence of a fully discrete scheme for two-dimensional neutron transport,SIAM J. Numer. Anal. 20, 951–966.

  11. Johnson, C. and Pitkäranta, J., 1986, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation,Math. Comp. 46, 1–26.

  12. Johnson, C. and Saranen, J., 1986, Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations,Math. Comp. 47, 1–18.

  13. Karamouzis, D. N., 1990, A space discontinuous Galerkin method for the one-dimensional unsteady convection-diffusion equation,Water Resour. Management 4, 175–185.

  14. Katopodes, N. D., 1984, A dissipative Galerkin scheme for open-channel flow,J. Hydraul. Engng., PASCE 110(4), 450–466.

  15. Katopodes, N., Wu Chien-Tai, and Karamouzis, D., 1984, Unified mass transport model for high and low groundwater dispersivities, Dept. of Civil Engineering, Univ. of Michigan, Ann Arbor, Michigan.

  16. Koussis, A. D., Saenz, M. A., and Tollis I. G., 1983, Pollution routing in streams,J. Hydraul. Engng., PASCE. 109, 1636–1651.

  17. Marchuk, G. I., 1986,Mathematical Models in Environmental Problems, Elsevier/North-Holland, Amsterdam.

  18. Nävert, Uno, 1982, A finite element method for convection-diffusion problems, PhD thesis, Chalmers Univ. of Technology, Göteborg.

  19. Raymond, W. H. and Garder, A., 1976, Selective damping in a Galerkin method for solving wave problems with variable grids,Monthly Weather Rev. 104, 1583–1590.

  20. Strang, G. and Fix, G. J., 1973,An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Karamouzis, D.N. A spacetime discontinuous Galerkin method for the two-dimensional unsteady convection-dispersion problems. Water Resour Manage 6, 35–45 (1992). https://doi.org/10.1007/BF00872186

Download citation

Key words

  • Artificial diffusion
  • convection-dispersion
  • Courant number
  • discontinuous Galerkin method
  • finite element computational scheme
  • mathematical model
  • Péclet number
  • Petrov-Galerkin
  • water contamination