A comparison of the GWCE and mixed P - P 1 formulations in finite-element linearized shallow-water models

International Journal for Numerical Methods in Fluids

Volumn 68, Issue 12, 2012, Pages 1497-1523

  Roux, D Y Le ^a   Walters, R ^b   Hanert, E ^c   Pietrzak, J ^d  

^a CNRS   (France)

^b Hydrodynamics   (Canada)

^c UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

^d DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Dispersion analysis; Finite elements; Generalized wave continuity equation; Gravity waves; Shallow water equations

Indexed keywords

ANALYTICAL RESULTS; COMMON STRATEGY; CONSERVATION PROPERTIES; DISPERSION ANALYSIS; EQUATION FORMULATION; FINITE ELEMENTS; FINITE-ELEMENT; GENERALIZED WAVE CONTINUITY EQUATIONS; NUMERICAL SOLUTION; PRESSURE MODES; PRIMITIVE EQUATIONS; PRIMITIVE VARIABLES; SHALLOW WATER EQUATIONS; SHALLOW-WATER; SHALLOW-WATER MODELS; TEST PROBLEM;

DATA STRUCTURES; DISPERSION (WAVES); EQUATIONS OF MOTION; GRAVITY WAVES;

FINITE ELEMENT METHOD;

EID: 84858709534     PISSN: 02712091     EISSN: 10970363     Source Type: Journal    
DOI: 10.1002/fld.2540     Document Type: Article

References (54)

1. Cushman-Roisin B. Introduction to Geophysical Fluid Dynamics. Prentice-Hall: Englewood Cliffs, NJ, 1994.
2. Pedlosky J. Geophysical Fluid Dynamics. Springer: Berl1987.
3. Walters RA, Carey GF. Analysis of spurious oscillation modes for the shallow water and Navier-Stokes equations. Computers and Fluids 1983; 11:51-68.
4. Arakawa A, Lamb VR. Computational design of the basic dynamical processes of the UCLA general circulation model. Methods in Computational Physics 1977; 17:173-265.
5. Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer: Berl1991.
6. Iskandarani M, Haidvogel D, Boyd J. A staggered spectral finite element model for the shallow-water equations. International Journal for Numerical Methods in Fluids 1995; 20:393-414.
7. Le Roux DY, Rostand V, Pouliot B. Analysis of numerically induced oscillations in 2D finite-element shallow-water models. Part I: inertia-gravity waves. SIAM Journal on Scientific Computing 2007; 29:331-360.
8. Batteen ML, Han YJ. On the computational noise of finite-difference schemes used in ocean models. Tellus 1981; 33:387-396.
9. Adcroft AJ, Hill CN, Marshall JC. A new treatment of the Coriolis terms in C-grid models at both high and low resolutions. Monthly Weather Review 1999; 127:1928-1936.
10. Walters RA, Carey GF. Numerical noise in ocean and estuarine models. Advances in Water Resources 1984; 7:15-20.
11. Rostand V, Le Roux DY, Carey G. Kernel analysis of the discretized finite difference and finite element shallow water models. SIAM Journal on Scientific Computing 2008; 31:531-556.
12. Kleptsova O, Pietrzak J, Stelling GS. On the accurate and stable reconstruction of tangential velocities in C-grid ocean models. Ocean Modelling 2009; 28:118-126.
13. Rostand V, Le Roux DY. Raviart-Thomas and Brezzi-Douglas-Marini finite element approximations of the shallow-water equations. International Journal for Numerical Methods in Fluids 2008; 57:951-976.
14. 1 finite-element pair in shallow-water models. SIAM Journal on Scientific Computing 2005; 27:394-414.
15. Le Roux DY, Carey GF. Stability/dispersion analysis of the discontinuous Galerkin linearized shallow-water system. International Journal for Numerical Methods in Fluids 2005; 48:325-347.
16. Randall DA. Geostrophic adjustment and the finite-difference shallow-water equations. Monthly Weather Review 1994; 122:1371-1377.
17. Williams RT, Zienkiewicz OC. Improved finite-element forms for the shallow-water wave equations. International Journal for Numerical Methods in Fluids 1981; 1:81-97.
18. King IP, Norton WR, Iceman KR. A finite element model for two-dimensional flow. In Finite Elements in Flow Problems, Oden JT (ed.). University of Alabama at Huntsville (UAH) Press, 1974; 133-137.
19. Raviart PA, Thomas JM. A mixed finite element method for 2nd order elliptic problems. Mathematical Aspects of the Finite Element Method. Lecture Notes in Mathematics.Springer: Berl1977.
20. Sigurdsson S. Treatment of the convective term in staggered finite element schemes for shallow water flow. In Computational Methods in Water Resources IX, Volume 1: Numerical Methods in Water Resources, Russel TF, Ewing RE, Brebia CA, Gray WG, Pinder GF (eds). Computational Mechanics Publications, Elsevier Applied Science, 1992; 291-298.
21. Williams RT. On the formulation of the finite-element prediction models. Monthly Weather Review 1981; 109:463-466.
22. Hua BL, Thomasset F. A noise-free finite element scheme for the two-layer shallow water equations. Tellus 1984; 36A:157-165.
23. Le Roux DY. A new triangular finite-element with optimum constraint ratio for compressible fluids. SIAM Journal on Scientific Computing 2001; 23:66-80.
24. Hughes TJR, Franca LP, Balestra M. A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accomodating equal-order interpolations. Computer Methods in Applied Mechanics and Engineering 1986; 59:85-99.
25. Cullen MJP, Hall CD. Forecasting and general circulation results from finite-element models. Quarterly Journal of the Royal Meteorological Society 1979; 105:571-592.
26. Staniforth A, Mitchell HL. A semi-implicit finite-element barotropic model. Monthly Weather Review 1977; 105:154-169.
27. Lynch DR, Gray WG. A wave equation model for finite element tidal computations. Computers and Fluids 1979; 7:207-228.
28. Kinnmark IPE, Gray WG. A two-dimensional analysis of the wave equation model for finite element tidal computations. International Journal for Numerical Methods in Engineering 1984; 20:369-383.
29. Luettich RA, Westerink JJ, Scheffner NW. An advanced three-dimensional circulation model of shelves, coasts, and estuaries. Report 1: Theory and Methodology of ADCIRC-2DDI and ADCIRC-3DL, Technical Report DRP-92-6, Department of the Army, Vicksburg, MS, 1992.
30. Dawson C, Westerink JJ, Feyen JC, Pothina D. Continuous, discontinuous and coupled discontinuous continuous Galerkin finite element methods for the shallow water equations. International Journal for Numerical Methods in Fluids 2006; 52:63-88.
31. Westerink JJ, Luettich RA, Feyen JC, Atkinson JH, Dawson C, Roberts HJ, Powell MD, Dunion JP, Kubatko EJ, Pourtaheri H. A basin- to channel-scale unstructured grid hurricane storm surge model applied to Southern Louisiana. Monthly Weather Review 2008; 136:833-864.
32. Le Bars Y, Lyard F, Jeandel C, Dardengo L. The AMANDES tidal model for the Amazon estuary and shelf. Ocean Modelling 2010; 31:132-149.
33. Kinnmark IPE. In The Shallow-water Wave Equations: Formulation, Analysis and Application, Brebbia CA, Orszag SA (eds), Lecture Notes in Engineering, vol. 15. Springer: Berl1986; 1-187.
34. 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; 23:523-538.
35. Kolar RL, Westerink JJ. A look back at 20 years of GWC-based shallow-water models. In Proceedings of the XIII International Conference on Computational Methods in Water Resources, Calgary, Canada, Bentley LR et al. (eds), vol. 2. 2000; 899-906.
36. Comblen R, Lambrechts J, Remacle JF, Legat V. Practical evaluation of five partly discontinuous finite element pairs for the non-conservative shallow water equations. International Journal for Numerical Methods in Fluids 2010; 63:701-724.
37. Le Roux DY, Pouliot B. Analysis of numerically induced oscillations in two-dimensional finite-element shallow-water models. Part II: free planetary waves. SIAM Journal on Scientific Computing 2008; 30:1971-1991.
38. 0. Ocean Modelling 2009; 28:24-33.
39. Le Roux DY, Hanert E, Rostand V, Pouliot B. Impact of mass lumping on gravity and Rossby waves in 2D finite-element shallow-water models. International Journal for Numerical Methods in Fluids 2009; 59:767-790.
40. Foreman MGG. An analysis of the wave equation model for finite element tidal computations. Journal of Computational Physics 1983; 52:290-312.
41. Kinnmark IPE, Gray WG. Stability and accuracy of spatial approximations for wave equation tidal models. Journal of Computational Physics 1985; 60:447-466.
42. Foreman MGG. A two-dimensional dispersion analysis of selected methods for solving the linearized shallow water equations. Journal of Computational Physics 1984; 56:287-323.
43. Gray WG, Lynch DR. Time-stepping schemes for finite element tidal model computations. Advances in Water Resources 1977; 1:83-95.
44. Atkinson JH, Westerink JJ, Luettich RA. Two-dimensional dispersion analyses of finite element approximations to the shallow water equations. International Journal for Numerical Methods in Fluids 2004; 45:715-749.
45. LeBlond PH, Mysak LA. Waves in the Ocean. Elsevier: Amsterdam, 1978; 602.
46. Walters RA, Barragy EJ. Comparison of h and p finite element approximations of the shallow water equations. International Journal for Numerical Methods in Fluids 1997; 24:61-79.
47. Thomasset F. Implementation of Finite Element Methods for Navier-Stokes Equations. Springer: Berl1981.
48. Porter B. Stability Criteria for Linear Dynamical Systems. Academic Press: New York, 1968.
49. Leendertse JJ. Aspects of a computational model for long period water-wave propagation. Technical Report RM-5294-PR, Sancta Monica, Rand Memorandum, 1967.
50. Hanert E, Le Roux DY, Legat V, Deleersnijder E. Advection schemes for unstructured grid ocean modelling. Ocean Modelling 2004; 7:39-58.
51. Henry RF, Duncalf DS, Walters RS, Osborne MJ, Murty TS. A study of tides and storm surges in offshore waters of the Meghna estuary using a finite element model. Mausam 1997; 48:519-530.
52. Dresback KM, Kolar RL, Dietrich JC. A 2D implicit time-marching algorithm for shallow water models based on the generalized wave continuity equation. International Journal for Numerical Methods in Fluids 2004; 45:253-274.
53. Hanert E, Le Roux DY, Legat V, Deleersnijder E. An efficient Eulerian finite-element method for the shallow-water equations. Ocean Modelling 2005; 10:115-136.
54. Walters RA, Lane EM, Henry RF. Semi-Lagrangian methods for a finite element coastal ocean model. Ocean Modelling 2007; 19:112-124.