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Reseach Article

Simulation of the Lock-Exchange Hydraulics using the Discontinuous Galerkin Method

by Nouh Izem, Mohammed Seaid, Mohamed Wakrim
International Journal of Computer Applications
Foundation of Computer Science (FCS), NY, USA
Volume 43 - Number 6
Year of Publication: 2012
Authors: Nouh Izem, Mohammed Seaid, Mohamed Wakrim
10.5120/6108-8325

Nouh Izem, Mohammed Seaid, Mohamed Wakrim . Simulation of the Lock-Exchange Hydraulics using the Discontinuous Galerkin Method. International Journal of Computer Applications. 43, 6 ( April 2012), 20-28. DOI=10.5120/6108-8325

@article{ 10.5120/6108-8325,
author = { Nouh Izem, Mohammed Seaid, Mohamed Wakrim },
title = { Simulation of the Lock-Exchange Hydraulics using the Discontinuous Galerkin Method },
journal = { International Journal of Computer Applications },
issue_date = { April 2012 },
volume = { 43 },
number = { 6 },
month = { April },
year = { 2012 },
issn = { 0975-8887 },
pages = { 20-28 },
numpages = {9},
url = { https://ijcaonline.org/archives/volume43/number6/6108-8325/ },
doi = { 10.5120/6108-8325 },
publisher = {Foundation of Computer Science (FCS), NY, USA},
address = {New York, USA}
}
%0 Journal Article
%1 2024-02-06T20:32:43.499927+05:30
%A Nouh Izem
%A Mohammed Seaid
%A Mohamed Wakrim
%T Simulation of the Lock-Exchange Hydraulics using the Discontinuous Galerkin Method
%J International Journal of Computer Applications
%@ 0975-8887
%V 43
%N 6
%P 20-28
%D 2012
%I Foundation of Computer Science (FCS), NY, USA
Abstract

Numerical simulations of the lock-exchange hydraulics have been carried out using a discontinuous Galerkin finite element method. The basic water circulation in the lock-exchange hydraulics consists in an upper layer of cold, fresh surface water and an opposite deep current of warmer, salty outflowing water. The governing equations are the well-established two-layer shallow water system including bathymetric forces. The considered discontinuous Galerkin method is a stable, highly accurate and locally conservative finite element method whose approximate solutions are discontinuous across interelement boundaries; this property renders the method ideally suited for the hp-adaptivity. The proposed method can handle complex topography using unstructured grids and it satisfies the conservation property. Several numerical results are presented to demonstrate the high resolution of the proposed method and to confirm its capability to provide accurate and efficient simulations for the lock-exchange hydraulics.

References
  1. R. Abgrall, S. Karni, Two-layer shallow water systems: a relaxation approach, SIAM J. Sci. Comput. 31 (2009) 1603–1627.
  2. V. Aizinger, C. Dawson, A discontinuous Galerkin method for two-dimensional flow and transport in shallow water, Advances in Water Resources 25 (2002) 67–84.
  3. Y. Bazilevs, T. Hughes, Weak imposition of Dirichlet boundary conditions in fluid mechanics, Computers & Fluids, in press 36 (2007) 12–26.
  4. F. Bouchut, T. Morales, An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment, M2AN Math. Model. Numer. Anal. 42 (2008) 683–698.
  5. M. Castro, J. García -Rodriguez, J. González -Vida, J. Macias, C. Parés, M. Vázquez-Cendón, Numerical simulation of two-layer shallow water flows through channels with irregular geometry, J. Comp. Physics. 195 (2004) 202–235.
  6. B. Cockburn, G. E. Karniadakis, C. W. S. (eds. ), Discontinuous Galerkin methods. Theory, computation and applications, Lecture Notes in Computational Science and Engineering, 11. Springer-Verlag, Berlin, 2000.
  7. B. Cockburn, C. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multi-dimensional systems, J. Comput. Phys. 141 (1998) 199–224.
  8. B. Cockburn, C. W. Shu, The Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, Journal of Scientific Computing. 16 (2001) 173–261.
  9. M. Dubiner, Spectral methods on triangles and other domains, Journal of Scientific Computing 6 (1991) 345–390.
  10. M. Dudzinski, M. Medvidova, Well-balanced path-consistent finite volume EG schemes for the two-layer shallow water equations, Computational Science and High Performance Computing. IV (2009) 121–136.
  11. C. Eskilsson, S. J. Sherwin, A triangular spectral/hp discontinuous Galerkin method for modelling 2d shallow water equations, Int. J. Numer. Methods Fluids. 45 (2004) 605–623.
  12. D. Farmer, L. Armi, Maximal two-layer exchange over a sill and through a combination of a sill and contraction with barotropic flow, J. Fluid Mech. 164 (1986) 53–76.
  13. T. W. F. X. Giraldo, A nodal triangle-based spectral element method for the shallow water equations on the sphere, Journal of Computational Physics 207 (2005) 129–150.
  14. F. X. Giraldo, T. Warburton, A high-order triangular discontinuous Galerkin oceanic shallow water model, Int. J. Numer. Meth. Fluids 56 (2008) 899–925.
  15. J. Hesthaven, From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM J. Numer. Anal. 35 (1998) 655–676.
  16. J. Hesthaven, T. Warburton, High-order nodal methods on unstructured grids. I. Time-domain solution of maxwells equations, J Comp Phys 181(1) (2002) 186–221.
  17. T. Hughes, G. Scovazzi, P. Bochev, A. Buffa, A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method, Computer Methods in Applied Mechanics and Engineering 195 (2006) 2761–2787.
  18. C. W. S. J. Qiu, B. C. Khoo, A numerical study for the performance of the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes, J. Comput. Phys. 212 (2006) 540–565.
  19. S. G. J. S. Hesthaven, D. Gottlieb, Spectral Methods for Time-Dependent Problems, Cambridge University Press, Cambridge, 2006.
  20. T. Koornwinder, Two-variable analogues of the classical orthogonal polynomials, in: R. A. Askey (Ed. ), Theory and Applications of Special Functions, Academic Press, San Diego, 1975.
  21. E. Kubatko, J. Westerink, C. Dawson, A. Buffa, hp discontinuous Galerkin methods for advection dominated problems in shallow water flow, Computer Methods in Applied Mechanics and Engineering 196 (2006) 437–451.
  22. A. Kurganov, G. Petrova, Central-upwind schemes for two-layer shallow water equations, SIAM J. Sci. Comput. 31 (2009) 1742–1773.
  23. W. Lee, A. Borthwick, P. Taylor, A fast adaptive quadtree scheme for a two-layer shallow water model, J. Comp. Physics. 230 (2011) 4848–4870.
  24. R. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.
  25. R. V. M. A. Taylor, B. A. Wingate, An algorithm for computing fekete points in the triangle, SIAM J. Numer. Anal. 38 (2000) 1707–1720.
  26. J. Macías, C. Parés, M. Castro, Improvement and generalization of a finite element shallow water solver to multi-layer systems, Int. J. Num. Methods Fluids. 31 (1999) 1037–1059.
  27. J. Proriol, Sur une famille de polynomes à deux variables orthogonaux dans un triangle, C. R. Acadamic Science, Paris 257, 1957.
  28. C. Shu, Total variation diminishing time discretizations, SIAM J. Sci. Stat. Comput. 9 (1988) 1073–1084.
  29. S. Tu, S. Allibadi, A slope limiting procedure in discontinuous Galerkin finite element method for gas dynamics applications, International Journal of Numerical Analysis and Modeling. 2 (2005) 163–178.
  30. C. Vreugdenhil, Two-layer shallow-water flow in two dimensions, a numerical study, J. Comp. Physics. 33 (1979) 169–184.
Index Terms

Computer Science
Information Sciences

Keywords

Discontinuous Galerkin Method Two-layer Shallow Water Equations Finite Element Lock-exchange Hydraulics