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High-order h-adaptive discontinuous Galerkin methods for ocean modelling

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Bernard, Paul-Emile [UCL] Chevaugeon, Nicolas Legat, Vincent [UCL] Deleersnijder, Eric [UCL] Remacle, Jean-François [UCL]

In this paper, we present an h-adaptive discontinuous Galerkin formulation of the shallow water equations. For a discontinuous Galerkin scheme using polynomials up to order p, the spatial error of discretization of the method can be shown to be of the order of h(p+1), where h is the mesh spacing. It can be shown by rigorous error analysis that the discontinuous Galerkin method discretization error can be related to the amplitude of the inter-element, jumps. Therefore, we use the information contained in jumps to build error metrics and size field. Results are presented for ocean modelling problems. A first experiment shows that the theoretical convergence rate is reached with the discontinuous Galerkin high-order h-adaptive method applied to the Stommel wind-driven gyre. A second experiment shows the propagation of an anticyclonic eddy in the Gulf of Mexico.

Document type Article de périodique (Journal article) – Article de recherche
Access type Accès restreint
Publication date 2007
Language Anglais
Journal information "Ocean Dynamics : theoretical, computational oceanography and monitoring" - Vol. 57, no. 2, p. 109-121 (2007)
Peer reviewed yes
Publisher Springer Heidelberg (Heidelberg)
issn 1616-7341
e-issn 1616-7228
Publication status Publié
Affiliations UCL - FSA/MECA - Département de mécanique
UCL - FSA/AUCE - Département d'architecture, d'urbanisme et de génie civil environnemental
Keywords shallow water equations H-adaptivity ; discontinuous Galerkin ; a posteriori error estimation
Links
  1. Adjerid S, Devine KD, Flaherty JE, Krivodonova L (2002) A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput Methods Appl Mech Eng 191:1097–1112
  2. Ainsworth M (2004) Dispersive and dissipative behavior of high order discontinuous Galerkin finite element methods. J Comput Phys 198(1):106–130
  3. Atkins HL, Shu C-W (1998) Quadrature-free implementation of discontinuous Galerkin methods for hyperbolic equations. AIAA J 36(5):775–782
  4. Baker TJ (1997) Mesh adaptation strategies for problems in fluid dynamics. Finite Elem Anal Des 25(3–4):243–273
  5. Bassi F, Rebay S (1997) A high-order accurate discontinuous finite element solution of the 2d Euler equations. J Comput Phys 138:251–285
  6. Behrens J (1998) Atmospheric and ocean modeling with an adaptive finite element solver for the shallow-water equations. Appl Numer Math 26:217–226
  7. Busalacchi AJ, O’Brian JJ (1980) The seasonal variability in a model of the tropic pacific. J Phys Oceanogr 10:1929–1951
  8. Chevaugeon N, Hillewaert K, Gallez X, Ploumhans P, Remacle J-F (2005a) Optimal numerical parameterization of discontinuous Galerkin method applied to wave propagation problems. J Comput Phys (in press)
  9. Chevaugeon N, Remacle J-F, Gallez X, Ploumans P, Caro S (2005b) Efficient discontinuous Galerkin methods for solving acoustic problems. In: 11th AIAA/CEAS Aeroacoustics Conference
  10. Chevaugeon N, Xin J, Hu P, Li X, Cler D, Flahertyand JE, Shephard MS (2005c) Discontinuous Galerkin methods applied to shock and blast problems. J Sci Comput 22(1):227–243
  11. Cockburn B, Karniadakis GE, Shu C-W (2000) Discontinuous Galerkin methods. Lect Notes Comput Sci Eng. Springer, Berlin Heidelberg New York
  12. Dawson C, Proft J (2002) Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations. Comput Methods Appl Mech Eng 191(41–42):4721–4746
  13. Dawson C, Proft J (2004) Coupled discontinuous and continuous Galerkin finite element methods for the depth-integrated shallow water equations. Comput Methods Appl Mech Eng 193(3–5)
  14. George PL, Borouchaki H, Laug P (2002) An efficient algorithm for 3d adaptive meshing. Adv Eng Softw 33(7–10):377–387, ISSN 0965-9978. DOI 10.1016/S0965-9978(02)00065-0
  15. Giraldo FX, Hesthaven JS, Warburton T (2002) Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations. J Comput Phys 181:499–525
  16. Hanert E, Le Roux DY, Legat V, Deleersnijder E (2004) Advection schemes for unstructured grid ocean modelling. Ocean Model 7:39–58
  17. Hanert E, Le Roux DY, Legat V, Deleersnijder E (2005) An efficient finite element method for the shallow water equations. Ocean Model 10:115–136
  18. Heinze T, Hense A (2002) The shallow water equations on the sphere and their lagrange–Galerkin-solution. Meteorol Atmos Phys 81:129–137
  19. Li X (2003) Mesh modification procedure for general 3-D non-manifold domains. PhD thesis, Renselear Polytechnic Institute
  20. Li X, Remacle J-F, Chevaugeon N, Shephard MS (2004) Anisotropic mesh gradation control. In 13th International Meshing Roundtable
  21. Lockard DP, Atkins HL (1999) Efficient implementations of the quadrature-free discontinuous Galerkin method. In Proceeding of 14th AIAA CFD conference. AIAA
  22. Luthar ME, O’Brien JJ (1985) A model of the seasonal circulation in the Arabian sea forced by observed winds. Progr Oceanogr 14:353–385
  23. Marchandise E, Chevaugeon N, Remacle J-F (2005) Spatial and spectral superconvergence of discontinuous Galerkin methods applied to hyperbolic problems. J Comput Appl Math, accepted for publication
  24. Nair RD, Thomas SJ, Loft RD (2005) A discontinuous Galerkin global shallow water model. Mon Weather Rev 133:876–888
  25. Naithani J, Deleersnijder E, Plisnier P-D (2003) Analysis of wind-induced thermocline oscillations of lake tanganyika. Environ Fluid Mech 3:23–39
  26. Pain CC, Piggott MD, Goddard AJH, Fang F, Gorman GJ, Marshall DP, Eaton MD, Power PW, de Oliveira CRE (2005) Three-dimensional unstructured mesh ocean modelling. Ocean Model 10:5–33
  27. Pietrzak J, Deleersnijder E, Schroeter J (eds) (2005) The second international workshop on unstructured mesh numerical modelling of coastal, shelf and ocean flows (delft, the Netherlands, September 23–25, 2003). Ocean Model (special issue) 10:1–252
  28. Remacle J-F, Li X, Shephard MS, Flaherty JE (2005) Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods. Int J Numer Methods Eng 62(7):899–923
  29. Remacle J-F, Soares Frazão S, Li X, Shephard MS (2006) An adaptive discretization of shallow-water equations based on discontinuous Galerkin methods. Int J Numer Methods Fluids 52:903–923
  30. Roe PL (1981) Approximate Riemann solvers, parameter vectors and difference schemes. J Comput Phys 43:357–372
  31. Schwanenberg D, Kongeter J (2000) A discontinuous Galerkin method for the shallow water equations with source terms. Comput Sci Eng 11:419–424
  32. Soares Frazão S, Zech Y (2002a) Undular bores and secondary waves-experiments and hybrid finite-volume modelling. J Hydraulic Res 40:33–34
  33. Soares Frazão S, Zech Y (2002b) Dam-break in channels with 90-degree bend. J Hydraul Res, American Society of Civil Engineers 128:956–968
  34. Speares W, Berzins M (1997) A 3d unstructured mesh adaptation algorithm for the time-dependent shock-dominated problems. Int J Numer Methods Fluids 25(1):81–104
  35. Stommel Henry, The westward intensification of wind-driven ocean currents, 10.1029/tr029i002p00202
  36. Vreugdenhill CB (1994) Numerical methods for shallow water flow. Water Science and Technologies Library. Kluwer
  37. Woodberry KE, Luther ME, O’Brien JJ (1989) The wind-driven seasonal circulation in the southern tropical Indian ocean. J Geophys Res 94:17,985–18,002
Bibliographic reference Bernard, Paul-Emile ; Chevaugeon, Nicolas ; Legat, Vincent ; Deleersnijder, Eric ; Remacle, Jean-François. High-order h-adaptive discontinuous Galerkin methods for ocean modelling. In: Ocean Dynamics : theoretical, computational oceanography and monitoring, Vol. 57, no. 2, p. 109-121 (2007)
Permanent URL http://hdl.handle.net/2078.1/37675