Coupled discontinuous and continuous Galerkin finite element methods for the depth-integrated shallow water equations

Computer Methods in Applied Mechanics and Engineering

Volumn 193, Issue 3-5, 2004, Pages 289-318

  Dawson, Clint ^a   Proft, Jennifer ^b,c  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b UNIVERSITÉ PARIS EST   (France)

^c CNRS   (France)

Author keywords

Coupled method; Discontinuous Galerkin method; Galerkin finite element method; Shallow water equations

Indexed keywords

ERROR ANALYSIS; ESTIMATION; FINITE ELEMENT METHOD; GALERKIN METHODS; WATER;

SHALLOW WATER EQUATIONS;

NONLINEAR EQUATIONS;

COUPLED ELEMENT METHOD; ERROR; FINITE ELEMENT METHOD; MATHEMATICAL MODELING; SHALLOW WATER; WAVE PROPAGATION;

EID: 0347930482     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2003.09.011     Document Type: Article

References (45)

1. Aizinger V. Dawson C.N. Discontinuous Galerkin methods for two-dimensional flow and transport in shallow water Adv. Water Resour. 2002 67-84
2. Alcrudo F. Garcia-Navarro P. A high-resolution Godunov-type scheme in finite volumes for the 2D shallow-water equations Int. J. Numer. Methods Fluids 16 1993 489-505
3. Baumann C.E. Oden J.T. A discontinuous hp finite element method for convection-diffusion problems Comput. Methods Appl. Mech. Engrg. 175 1999 311-341
4. Bey K.S. Oden J.T. hp-version discontinuous Galerkin methods for hyperbolic conservation laws Comput. Methods Appl. Mech. Engrg. 133 1996 259-286
5. Blain C.A. Westerink J. Luettich R. Grid convergence studies for the prediction of hurricane storm surge Int. J. Numer. Methods Fluids 26 1998 1-33
6. Blain C.A. Westerink J.J. Luettich R.A. The influence of domain size on the response characteristics of a hurricane storm surge model J. Geophys. Res. 99 C9 1994 18467-18479
7. Brenner S. Scott L.R. The Mathematical Theory of Finite Element Methods 1994 Springer Verlag New York
8. Castillo P. Cockburn B. Perugia I. Schötzau D. An a priori error analysis of the local discontinuous Galerkin method for elliptic problems SIAM J. Numer. Anal. 38 2000 1676-1706
9. Chippada S. Dawson C.N. Martinez M. Wheeler M. Finite element approximations to the System of shallow water equations, Part I: Continuons time a priori error estimates SIAM J. Numer. Anal. 35 1995 692-711
10. Chippada S. Dawson C.N. Martinez M. Wheeler M. Finite element approximations to the System of shallow water equations, Part II: Discrete time a priori error estimates SIAM J. Numer. Anal. 36 1996 226-250
11. Chippada S. Dawson C.N. Martinez M. Wheeler M. A Godunov-type finite volume method for the System of shallow water equations Comput. Methods Appl. Mech. Engrg. 151 1998 105-129
12. B. Cockburn, C. Dawson, Some extensions of the local discontinuous Galerkin method for convection-diffusion equations in multidimensions, in: J.R. Whiteman (Ed.), The Proceedings of the Conference on the Mathematics of Finite Elements and Applications: MAFELAP X, Elsevier, 2000, pp. 225-238
13. B. Cockburn, G. Karniadakis, C.-W. Shu, The development of discontinuous Galerkin methods, in: B. Cock-burn, G.E. Karniadakis, C.-W. Shu (Eds.), First International Symposium on Discontinuous Galerkin Methods, Springer-Verlag, 2000, pp. 5-50
14. Cockburn B. Lin S.Y. Shu C.W. TVB RK local projection discontinuous Galerkin finite element for conservation laws III: One-dimensional systems J. Comp. Phys. 84 1989 90-113
15. Cockburn B. Shu C.W. TVB RK local projection discontinuous Galerkin finite element for conservation laws II: General framework Math. Comp. 52 1989 411-435
16. Cockburn B. Shu C.W. TVB RK local projection discontinuous Galerkin finite element for conservation laws IV: The multidimensional case Math. Comp. 54 1990 545-581
17. 1-discontinuous Galerkin method for scalar conservation laws RAIRO Model. Math. Anal. Numer. 25 1991 337-361
18. Cockburn B. Shu C.W. The local discontinuous Galerkin method for time dependent convection-diffusion systems SIAM J. Numer. Anal. 35 1998 2440-2463
19. Cockburn B. Shu C.W. TVB RK local projection discontinuous Galerkin finite element for conservation laws V: Multidimensional systems J. Comput. Phys. 141 1998 199-224
20. Dawson C. Proft J. A priori error estimates for interior penalty versions of the local discontinuous Galerkin method applied to transport equations Numer. Methods Partial Differential Equations 17 2001 545-564
21. C. Dawson, S. Sun, M.F. Wheeler, Compatible algorithms for coupled flow and transport, submitted for publication
22. Dawson C.N. Conservative, shock-capturing transport methods with nonconservative velocity approximations Comput. Geosci. 3 1999 205-227
23. Dawson C.N. Martinez-Canales M. A characteristics-Galerkin approximation to a system of shallow water equations Numer. Math. 86 2001 239-256
24. Dawson C.N. Proft J. Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations Comput. Methods Appl. Mech. Engrg. 191 2002 4721-4746
25. Dawson C.N. Proft J. Coupling of continuous and discontinuous Galerkin methods for transport problems Comput. Methods Appl. Mech. Engrg. 191 2002 3213-3231
26. Dawson C.N. Proft J. Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations Comput. Methods Appl. Mech. Engrg. 191 2002 4721-4746
27. Foremann M.G.G. An analysis of the wave equation model for finite element tidal computations Comput. Phys. 52 1983 290-312
28. M.G.G. Foremann, A comparison of tidal models for the Southwest coast of Vancouver Island, Computational Methods in Water Resources: Proceedings of the VII International Conference, MIT, 1988
29. Gray W.G. A finite element study of tidal flow data for the North Sea and English Channel Adv. Water Resour. 12 1989 143-154
30. Gray W.G. Drolet J. Kinnmark P.E. A simulation of tidal flow in the southern part of the North Sea and the English Channel Adv. Water Resour. 10 1987 131-137
31. Kinnmark I. The Shallow Water Wave Equations: Formulation, Analysis and Application 1985 Springer-Verlag
32. Kolar R.L. Westerink J. Cantekin M. Blain C. Aspects of nonlinear simulations using shallow water models based on the wave continuity equation Comput. Fluids 23 1994 523-538
33. LeSaint P. Raviart P.A. On a finite element method for solving the neutron transport equations Mathematical Aspects Finite Elements in Partial Differential Equations 1974 Academic Press
34. LeVeque R.J. Numerical Methods for Conservation Laws 1992 Birkhauser Basel
35. R.A. Luettich, J.J. Westerink, N.W. Scheffner, ADCIRC: An advanced three-dimensional circulation model for shelves, coasts and estuaries, tech. rep., US Army Corps of Engineers, Washington, DC, 1992
36. Lynch D.R. Gray W.G. A wave equation model for finite element tidal compilations Comput. Fluids 7 1979 207-228
37. Oden J.T. Babuška I. Baumann C.E. A discontinuous hp finite element method for diffusion problems J. Comput. Phys. 146 1998 491-519
38. B. Riviere, M.F. Wheeler, V. Girault, A discontinuous Galerkin method with non-overlapping domain decomposition for Stokes and Navier-Stokes problems, Mathematics of Computation, accepted
39. Riviere B. Wheeler M.F. Girault V. Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I Comput. Geosci. 8 1999 337-360
40. T. Weiyan, Shallow Water Hydrodynamics, Elsevier Oceanography Series, 55, Elsevier Amsterdam, 1992
41. Werner F.E. Lynch D.R. Field verification of wave equation tidal dynamics in the English Channel and southern North Sea Adv. Water Resour. 10 1987 115-130
42. Werner F.E. Lynch D.R. Harmonic structure of English Channel/Southern Bight tides from a wave equation simulation Adv. Water Resour. 12 1989 121-142
43. Westerink J. Luettich R. Baptista A. Scheffner N. Farrar P. Tide and storm surge predictions using a finite element model J. Hydraul. Engrg. 118 1992 1373-1390
44. J.J. Westerink, R.A. Luettich, R.L. Kolar, Advances in finite element modeling of coastal ocean hydrodyanmics, in: 11th International Conference on Computational Methods in Water Resources, Computational Mechanics Publications, Southampton, UK, 1996
45. Wheeler M.F. An elliptic collocation-finite element method with interior penalties SIAM J. Numer. Anal. 15 1978 152-161