Practical evaluation of five partly discontinuous finite element pairs for the non-conservative shallow water equations

International Journal for Numerical Methods in Fluids

Volumn 63, Issue 6, 2010, Pages 701-724

  Comblen, Richard ^a   Lambrechts, Jonathan ^a   Remacle, Jean François ^a   Legat, Vincent ^a  

^a UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

Author keywords

Convergence; Discontinuous Galerkin; Finite element; Non conforming element; Riemann solver; Shallow water equations

Indexed keywords

ASYMPTOTIC LIMITS; CONFORMING ELEMENTS; CORIOLIS; DISCONTINUOUS FINITE ELEMENT; DISCONTINUOUS GALERKIN; FINITE ELEMENT; INVISCID FLOWS; LINEAR ELEMENT; MIXED FORMULATIONS; NUMERICAL SCHEME; OPTIMAL RATE; RIEMANN SOLVER; SECOND ORDERS; SHALLOW WATER EQUATIONS; SHALLOW WATER FLOW; SPATIAL ACCURACY; TEST CASE; WEAK FORMULATION;

FINITE ELEMENT METHOD;

GALERKIN METHODS;

EID: 77953564917     PISSN: 02712091     EISSN: 10970363     Source Type: Journal    
DOI: 10.1002/fld.2094     Document Type: Article

References (44)

1. Mesinger F, Arakawa A. Numerical Methods Used in Atmospheric Models. GARP Publications Series No. 17, WMO - ICSU, 1976.
2. Griffies S. Fundamentals of Ocean Climate Models. Princeton University Press: Princeton, NJ, 2004.
3. Hartmann R, Houston P. Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations. Journal of Computational Physics 2002; 183:508-532.
4. Bernard PE, Chevaugeon N, Legat V, Deleersnijder E, Remacle JF. High-order h-adaptive discontinuous Galerkin methods for ocean modelling. Ocean Dynamics 2007; 57:109-121.
5. Burbeau A, Sagaut P. A dynamic p-adaptive discontinuous Galerkin method for viscous flow with shocks. Computer and Fluids 2005; 34:401-417.
6. Cockburn B, Shu CW. The Runge-Kutta discontinuous Galerkin method for conservation laws V-multidimensional systems. Journal of Computational Physics 1998; 141:191-224.
7. Chevaugeon N, Xin J, Hu P, Li X, Cler D, Flaherty JE, Shephard MS. Discontinuous Galerkin methods applied to shock and blast problems. Journal of Scientific Computing 2005; 22-23:227-243.
8. Remacle JF, Frazão SS, Li X, Shephard MS. An adaptive discretization of shallow-water equations based on discontinuous Galerkin methods. International Journal for Numerical Methods in Fluids 2006; 52:903-923.
9. Nair RD, Thomas SJ, Loft RD. A discontinuous Galerkin global shallow water model. Monthly Weather Review 2005; 133:876-888.
10. Giraldo FX. High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere. Journal of Computational Physics 2006; 214:447-465. DOI: 10.1016/j.jcp.2005.09.029.
11. Aizinger V, Dawson C. A discontinuous Galerkin method for two-dimensional flow and transport in shallow water. Advances in Water Resources 2002; 25:67-84.
12. Kubatko EJ, Westerink JJ, Dawson C. hp discontinuous Galerkin methods for advection dominated problems in shallow water flows. Computational Methods in Applied Mechanics and Engineering 2006; 196:437-451.
13. Bernard PE, Remacle JF, Legat V. Boundary discretization for high order discontinuous Galerkin computations of tidal flows around shallow water islands. International Journal for Numerical Methods in Fluids 2009; 59:535-557. DOI: 10.1002/fld.1831.
14. Schwanenberg D, Harms M. Discontinuous Galerkin finite-element method for transcritical two-dimensional shallow water flows. Journal of Hydraulic Engineering 2004; 130:412-421.
15. Arnold DN, Brezzi F, Cockburn B, Marini LD. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis 2002; 39:1749-1779.
16. Cockburn B, Shu C. The local discontinuous Galerkin finite element method for convection-diffusion systems. SIAM Journal on Numerical Analysis 1998; 35:2440-2463.
17. Riviere B. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, Frontiers in Mathematics, vol. 35. SIAM: Philadelphia, FL, 2008.
18. Donea J, Huerta A. Finite Element Methods for Flow Problem. Wiley: New York, 2003.
19. Bova SW, Carey GF. An entropy variable form and applications for the two-dimensional shallow water equations. International Journal for Numerical Methods in Fluids 1996; 23:29-46.
20. Hauke G. A symmetric form for computing transient shallow water flows. Computer Methods in Applied Mechanics and Engineering 1998; 163:111-122.
21. Ribeiro FLB, Galeão AC, Landau L. Edge-based finite element method or shallow water equations. International Journal for Numerical Methods in Fluids 2001; 36:659-685.
22. Hauke G. A stabilized finite element method for the Saint-Venant equations with application to irrigation. International Journal for Numerical Methods in Fluids 2002; 38:963-984.
23. Le Roux DY, Staniforth A, Lin CA. Finite elements for shallow-water equation ocean model. Monthly Weather Review 1998; 126:1931-1951.
24. Le Roux DY, Lin CA, Staniforth A. A semi-implicit semi-Lagrangian finite-element shallow-water ocean model. Monthly Weather Review 2000; 128:1384-1401.
25. Le Roux DY. A new triangular finite-element with optimum constraint ratio for compressible fluids. SIAM Journal on Scientific Computing 2001; 23:66-80.
26. Hanert E, Roux DYL, Legat V, Deleersnijder E. An efficient Eulerian finite element method for the shallow water equations. Ocean Modelling 2005; 10:115-136.
27. Le Roux DY, Sene A, Rostand V, Hanert E. On some spurious mode issues in shallow-water models using a linear algebra approach. Ocean Modelling 2005; 10:83-94.
28. 1 finite-element pair in shallow-water models. SIAM Journal on Scientific Computing 2005; 27:394-414.
29. 0. Ocean Modelling 2009; 28:24-33. DOI: 10.1016/j.ocemod.2008.07.002.
30. Hua BL, Thomasset F. A noise-free finite element scheme for the two-layer shallow water equations. Tellus 1984; 36A:157-165.
31. 1 pair. Ocean Modelling 2009; 28:2-11. DOI: 10.1016/j.ocemod.2008.03.005.
32. Cotter CJ, Ham DA, Pain CC. A mixed discontinuous/continuous finite element pair for shallow-water ocean modelling. Ocean Modelling 2009; 26:86-90. DOI: 10.1016/j.ocemod.2008.09.002.
33. Cotter CJ, Ham DA, Pain CC, Sebastian Reich S. LBB stability of a mixed Galerkin finite element pair for fluid flow simulations. Journal of Computational Physics 2009; 228:336-348. DOI: 10.1016/j.jcp.2008.09.014.
34. Shahbazi K. An explicit expression for the penalty parameter of the interior penalty method. Journal of Computational Physics 2005; 205:401-407.
35. LeVeque RJ. Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press: Cambridge, 2002.
36. Geuzaine C, Remacle JF. Gmsh: A 3-D finite element mesh generator with buit-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering 2009; Early View. DOI: 10.1002/nme.2579.
37. Brezzi F, Fortin M. Mixed and Hybrid Finite Element Methods. No. 15 in Springer Series in Computational Mathematics. Springer: Berlin, 1991.
38. Pedlosky J. Geophysical Fluid Dynamics (2nd edn). Springer: Heidelberg, 1987.
39. 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.
40. Le Roux DY, Rostand V, Pouliot B. Analysis of numerically induced oscillations in 2d finite-element shallow-water models part II: free planetary waves. SIAM Journal on Scientific Computing 2008; 30:1971-1991.
41. Bassi F, Crivellini A, Di Pietro DA, Rebay S. An artificial compressiblility flux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations. Journal of Computational Physics 2006; 218:794-815.
42. Di Pietro DA. Analysis of discontinuous Galerkin approximation of the Stokes problem based on an artificial compressibility flux. International Journal for Numerical Methods in Fluids 2007; 55:793-813.
43. Cockburn B, Kanschat G, Schötzau D, Schwab C. Local discontinuous Galerkin methods for the Stokes system. SIAM Journal on Numerical Analysis 2002; 40:319-343.
44. Comblen R, Legrand S, Deleersnijder E, Legat V. A finite element method for solving the shallow water equations on the sphere. Ocean Modelling 2009; 28:12-23. DOI: 10.1016/j.ocemod.2008.05.004.