Continuous, discontinuous and coupled discontinuous-continuous Galerkin finite element methods for the shallow water equations

International Journal for Numerical Methods in Fluids

Volumn 52, Issue 1, 2006, Pages 63-88

  Dawson, Clint ^a   Westerink, Joannes J ^b   Feyen, Jesse C ^b   Pothina, Dharhas ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b University of Notre Dame   (United States)

Author keywords

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

Indexed keywords

APPROXIMATION THEORY; FINITE ELEMENT METHOD; GALERKIN METHODS; NUMERICAL METHODS;

ADVECTIVE FLOWS; COUPLED FINITE ELEMENT METHOD; DISCONTINUOUS GALERKIN METHOD; SHALLOW WATER EQUATIONS;

FLOW OF WATER;

APPROXIMATION THEORY; FINITE ELEMENT METHOD; FLOW OF WATER; GALERKIN METHODS; NUMERICAL METHODS;

EID: 33747496679     PISSN: 02712091     EISSN: 10970363     Source Type: Journal    
DOI: 10.1002/fld.1156     Document Type: Article

References (50)

1. Weiyan T. Shallow Water Hydrodynamics, Elsevier Oceanography Series, vol. 55. Elsevier: Amsterdam, 1992.
2. Westerink JJ, Luettich RA, Baptista AM, Scheffner NW, Farrar P. Tide and storm surge predictions using a finite element model. Journal of Hydraulic Engineering 1992; 118:1373-1390.
3. Blain CA, Westerink JJ, Luettich RA. The influence of domain size on the response characteristics of a hurricane storm surge model. Journal of Geophysical Research 1994; 99(C9):18467-18479.
4. Lynch DR, Gray WG. A wave equation model for finite element computations. Computers and Fluids 1979; 7:207-228.
5. King IP, Norton WR. Recent application of RMA's finite element models for two-dimensional hydrodynamics and water quality. In Finite Elements in Water Resources II, Brebbia CA, Gray WG, Pinder GF (eds). Pentech Press: London, 1978.
6. Kawahara M, Hirano H, Tsuhora K, Iwagaki K. Selective lumping finite element method for shallow water equations. International Journal for Numerical Methods in Engineering 1982; 2:99-112.
7. Szymkiewicz R. Oscillation-free solution of shallow water equations for nonstaggered grid. Journal of Hydraulic Engineering 1993; 119:1118-1137.
8. Zienkiewicz OC, Ortiz P. A split-characteristic based finite element model for the shallow water equations. International Journal for Numerical Methods in Fluids 1995; 20:1061-1080.
9. Kinnmark IPE. The shallow water wave equations: formulation, analysis, and application. Ph.D. Thesis, Department of Civil Engineering, Princeton University, 1984.
10. Gray WG, Drolet J, Kinnmark IPE. A simulation of tidal flow in the southern part of the North Sea and the English Channel. Advances in Water Resources 1987; 10:131-137.
11. Gray WG. A finite element study of tidal flow data for the North Sea and English Channel. Advances in Water Resources 1989; 12:143-154.
12. Foreman MGG. An analysis of the wave equation model for finite element tidal computations. Computational Physics 1983; 52:290-312.
13. Foreman MGG. A comparison of tidal models for the southwest coast of Vancouver Island. In Computational Methods in Water Resources: Proceedings of the VII International Conference, MIT, Celia MA (ed.), vol. 2. Elsevier: Amsterdam, 1988.
14. Luettich RA, Westerink JJ, Scheffner NW. ADCIRC: an advanced three-dimensional circulation model for shelves, coasts and estuaries, Report 1: theory and methodology of ADCIRC-2DDI and ADCIRC-3DL. Dreding Research Program Technical Report DRP-92-6, U.S. Army Engineers Waterways Experiment Station, Vicksburg, MS. U.S. Army Corps of Engineers: Washington, DC, 1992.
15. Werner FE, Lynch DR. Field verification of wave equation tidal dynamics in the English Channel and southern North Sea. Advances in Water Resources 1987; 10:115-130.
16. Werner FE, Lynch DR. Harmonic structure of English Channel/Souther Bight tides from a wave equation simulation. Advances in Water Resources 1987; 12:121-142.
17. Blain CA, Westerink JJ, Luettich RA. Grid convergence studies for the prediction of hurricane storm surges. International Journal for Numerical Methods in Fluids 1998; 26:369-401.
18. Kolar RL, Westerink JJ, Cantekin ME, Blain CA. Aspects of nonlinear simulations using shallow water models based on the wave continuity equation. Computers and Fluids 1994; 23(3):523-538.
19. Westerink JJ, Luettich RA, Kolar RL. Advances in finite element modeling of coastal ocean hydrodynamics. In 11th International Conference on Computational Methods in Water Resources, Aldama A (ed.), vol. 2. Computational Mechanics Publications: Southampton, U.K., 1996.
20. Chippada S, Dawson CN, Martinez ML, Wheeler MF. Finite element approximations to the system of shallow water equations, Part I: Continuous time a priori error estimates. SIAM Journal on Numerical Analysis 1998; 35:692-711.
21. Chippada S, Dawson CN, Martinez ML, Wheeler MF. Finite element approximations to the system of shallow water equations, Part II: Discrete time a priori error estimates. SIAM Journal on Numerical Analysis 1998; 36(1):226-250.
22. Alcrudo F, Garcia-Navarro P. A high-resolution Godunov-type scheme in finite volumes for the 2d shallow-water equations. International Journal for Numerical Methods in Fluids 1993; 16:489-505.
23. Chippada S, Dawson CN, Martinez ML, Wheeler MF. A Godunov-type finite volume method for the system of shallow water equations. Computer Methods in Applied Mechanics and Engineering 1998; 151:105-129.
24. Aizinger V, Dawson C. Discontinuous Galerkin methods for two-dimensional flow and transport in shallow water. Advances in Water Resources 2002; 25:67-84.
25. Giraldo FX, Hesthaven JS, Warburton T. Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations. Journal of Computational Physics 2002; 181:499-525.
26. Schwanenberg D, Harms M. Discontinuous Galerkin finite element method for transcritical two-dimensional shallow water flows. Journal of Hydraulic Engineering 2004; 130(5):412-421.
27. Eskilsson C, Sherwin SJ. A triangular spectral/h p discontinuous Galerkin method for modeling 2D shallow water equations. International Journal for Numerical Methods in Fluids 2004; 45(6):605-623.
28. Dawson C. Conservative, shock-capturing transport methods with nonconservative velocity approximations. Computational Geosciences 1999; 3:205-227.
29. 1 -discontinuous Galerkin method for scalar conservation laws. RAIRO Modélisation Mathématique et Analyse Numérique 1991; 25:337-361.
30. Cockburn B, Shu CW. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: general framework. Mathematics of Computation 1989; 52:411-435.
31. Cockburn B, Lin SY, Shu CW. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. Journal of Computational Physics 1989; 84:90-113.
32. Cockburn B, Hou S, Shu CW. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Mathematics of Computation 1990; 54: 545-581.
33. Cockburn B, Shu CW. The Runge-Kutta discontinuous Galerkin finite element method for conservation laws V: multidimensional systems. Journal of Computational Physics 1998; 141:199-224.
34. Bey KS, Oden JT. hp-version discontinuous Galerkin methods for hyperbolic conservation laws. Computer Methods in Applied Mechanics and Engineering 1996; 133:259-286.
35. Baumann CE, Oden JT. A discontinuous h p finite element method for convection-diffusion problems. In Computer Methods in Applied Mechanics and Engineering special issue on Spectral, Spectral Element, and hp Methods in CFD, Ainsworth M, Karniadakis GE, Bernardi C (eds), vol. 175. 1999; 311-341.
36. Cockburn B, Shu CW. The local discontinuous Galerkin finite element method for convection-diffusion systems. SIAM Journal on Numerical Analysis 1998; 35:2440-2463.
37. Cockburn B, Dawson C. Some extensions of the local discontinuous Galerkin method for convection-diffusion equations in multidimensions. In Proceedings of the Conference on the Mathematics of Finite Elements and Applications: MAFELAP X, Whiteman JR (ed.). Elsevier: Amsterdam, 2000.
38. Dawson C, Proft J. A priori error estimates for interior penalty versions of the local discontinuous Galerkin method applied to transport equations. Numerical Methods for Partial Differential Equations 2001; 17(6): 545-564.
39. Wheeler MF. An elliptic collocation-finite element method with interior penalties. SIAM Journal on Numerical Analysis 1978; 15:152-161.
40. Baker GA. Finite element methods for elliptic equations using nonconforming elements. Mathematics of Computation 1977; 31:45-59.
41. Castillo P, Cockburn B, Perugia I, Schötzau D. An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM Journal on Numerical Analysis 2000; 38(5):1676-1706.
42. Oden JT, Babuška I, Baumann CE. A discontinuous h p finite element method for diffusion problems. Journal of Computational Physics 1998; 146:491-519.
43. Rivière B, Wheeler MF, Girault V. Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems, Part I. Computational Geosciences 1999; 3:337-360.
44. Cockburn B, Karniadakis GE, Shu CW. The development of discontinuous Galerkin methods. In First International Symposium on Discontinuous Galerkin Methods, Cockburn B, Kamiadakis GE, Shu CW (eds), Lecture Notes in Computational Science and Engineering, vol. 11. Springer: Berlin, 2000; 3-50.
45. Dawson C, Proft J. Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations. Computer Methods in Applied Mechanics and Engineering 2002; 191:4721-4746.
46. Blain CA, Massey TC. Application of a coupled discontinuous-continuous Galerkin finite element shallow water model to coastal ocean dynamics. Ocean Modelling 2005; 10(3-4):283-315.
47. Durlofsky L, Engquist B, Osher S. Triangle based adaptive stencils for the solution of hyperbolic conservation laws. Journal of Computational Physics 1992; 98:64-83.
48. Lynch DR, Gray WG. Analytical solutions for computer flow model testing. Journal of the Hydraulics Division (ASCE) 1978; HY-10:1409-1428.
49. Atkinson JH, Westerink JJ, Luettich RA. Two dimensional dispersion analysis of finite element approximations to the shallow water equations. International Journal for Numerical Methods in Fluids 2004; 45(7):715-749.
50. Roache PJ. Verification and Validation in Computational Science and Engineering. Hermosa Publishers: Albuquerque, 1998.