High-order discontinuous Galerkin schemes on general 2D manifolds applied to the shallow water equations

Journal of Computational Physics

Volumn 228, Issue 17, 2009, Pages 6514-6535

  Bernard, P E ^a   Remacle, J F ^a   Comblen, R ^a   Legat, V ^a   Hillewaert, K ^b  

^a UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

^b CENAERO   (Belgium)

Author keywords

Discontinuous Galerkin method; High order finite elements; Shallow water equations; Spherical geometry

Indexed keywords

EQUATIONS OF MOTION; FINITE ELEMENT METHOD; GALERKIN METHODS;

CONSERVATION LAW; DISCONTINOUS GALERKIN METHODS; DISCONTINUOUS GALERKIN SCHEMES; HIGH-ORDER; HIGH-ORDER FINITE ELEMENTS; HIGHER-ORDER; NEW APPROACHES; SHALLOW WATER EQUATIONS; SPHERICAL GEOMETRIES; VECTORIAL FIELDS;

ORTHOGONAL FUNCTIONS;

EID: 67650131339     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2009.05.046     Document Type: Article

References (62)

1. Adjerid S., Devine K.D., Flaherty J.E., and Krivodonova L. A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Computer Methods in Applied Mechanics and Engineering 191 (2002) 1097-1112
2. Ainsworth M. Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. Journal of Computational Physics 198 (2004) 106-130
3. Aizinger V., and Dawson C. A discontinuous Galerkin method for two-dimensional flow and transport in shallow water. Advances in Water Resources 25 (2002) 67-84
4. Aizinger V., and Dawson C. The local discontinuous Galerkin method for three-dimensional shallow water flow. Computer Methods in Applied Mechanics and Engineering 196 (2007) 734-746
5. Bassi F., and Rebay S. A high-order accurate discontinuous finite element solution of the 2D Euler equations. Journal of Computational Physics 138 (1997) 251-285
6. Bernard P.-E., Chevaugeon N., Legat V., Deleersnijder E., and Remacle J.-F. High-order h-adaptative discontinuous Galerkin methods for ocean modelling. Ocean Dynamics 57 (2007) 109-121
7. P.-E. Bernard, J.-F. Remacle, V. Legat, Boundary discretization for high order discontinuous Galerkin computations of tidal flows around shallow water islands, International Journal for Numerical Methods in Fluids, in press, doi:10.1002/fld.1831.
8. Bernard P.-E., Remacle J.-F., and Legat V. Dispersion analysis of discontinuous Galerkin schemes applied to Poincaré, Kelvin and Rossby waves. Journal of Scientific Computing 34 (2008) 26-47
9. K. Bey, J. Oden, A Runge-Kutta discontinuous Galerkin finite element method for high speed flows, in: Proceedings of AIAA 10th Computational Fluid Dynamics Conference, Honolulu, Hawaii, 1991.
10. Biswas R., Devine K., and Flaherty J. Parallel, adaptive finite element methods for conservation laws. Applied Numerical Mathematics 14 (1994) 255-283
11. N. Chevaugeon, J.-F. Remacle, X. Gallez, P. Ploumans, S. Caro, Efficient discontinuous Galerkin methods for solving acoustic problems, in: Proceedings of the 11th AIAA/CEAS Aeroacoustics Conference, 2005.
12. Chevaugeon N., Xin J., Hu P., Li X., Cler D., Flaherty J.E., and Shephard M.S. Discontinuous Galerkin methods applied to shock and blast problems. Journal of Scientific Computing 22-23 (2005) 227-243
13. Cho J.Y.-K., and Polvani L.M. The emergence of jets and vortices in freely evolving, shallow-water turbulence on a sphere. Physics of Fluids 8 (1996) 1531-1552
14. B. Cockburn, G.E. Karniadakis, C.W. Shu, Discontinuous Galerkin Methods. Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, vol. 11, Springer-Verlag, Berlin, 2000.
15. Cockburn B., and Shu C.-W. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. Journal of Scientific Computing 16 (2001) 173-261
16. Comblen R., Legrand S., Deleersnijder E., and Legat V. A finite element method for solving the shallow water equations on the sphere. Ocean Modelling 28 (2009) 12-23
17. Côté J. A Lagrange multiplier approach for the metric terms of semi-Lagrangian models on the sphere. Quarterly Journal of the Royal Meteorological Society 114 (1988) 1347-1352
18. Dawson C., and Proft J. Coupled discontinuous and continuous Galerkin finite element methods for the depth-integrated shallow water equations. Computer Methods in Applied Mechanics and Engineering 193 (2004) 289-318
19. K. Devine, J. Flaherty, S. Wheat, A. Maccabe, A massively parallel adaptive finite element method with dynamic load balancing, in: Proceedings of the Supercomputing - SAND 93-0936C, IEEE Computer Society Press, 1993, pp. 2-11.
20. Eskilsson C., and Sherwin S. A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations. International Journal for Numerical Method in Fluids 45 (2004) 605-623
21. Dupont F., and Lin C.A. The adaptive spectral element method and comparisons with more traditional formulations for ocean modeling. Journal of Atmospheric and Oceanic Technology 21 (2004) 135-147
22. J. Font, Numerical hydrodynamics in general relativity, Living Reviews in Relativity (2000).
23. Giraldo F.X., and Restelli M. A study of spectral element and discontinuous Galerkin methods for mesoscale atmospheric models: equation sets and test cases. Journal of Computational Physics 227 (2008) 3849-3877
24. Giraldo F.X., and Warburton T. A high-order triangular discontinuous Galerkin oceanic shallow water model. International Journal for Numerical Methods in Fluids 56 (2008) 899-925
25. Geuzaine C., and Remacle J.-F. Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering (2009) 10.1002/nme.2579
26. Gilman P. Magnetohydrodynamics shallow water equations for the solar tachocline. Astrophysical Journal 544 (2000) L79-L82
27. Giraldo F.X. Lagrange-Galerkin methods on spherical geodesic grids: the shallow water equations. Journal of Computational Physics 160 (2000) 336-368
28. Giraldo F.X. High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere. Journal of Computational Physics 214 (2006) 447-465
29. Giraldo F.X., Hesthaven J.S., and Warburton T. Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations. Journal of Computational Physics 181 (2002) 499-525
30. Giraldo F.X., and Warburton T. A nodal triangle-based spectral element method for the shallow water equations on the sphere. Journal of Computational Physics 207 (2005) 129-150
31. Hesthaven J.S. From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM Journal of Numerical Analysis 35 (1998) 655-676
32. Hillewaert K., Chevaugeon N., Geuzaine P., and Remacle J.-F. Hierarchic multigrid iteration strategy for the discontinuous Galerkin solution of the steady Euler equations. International Journal for Numerical Methods in Fluids 51 (2006) 1157-1176
33. Iacono R., Struglia M., and Ronchi C. Spontaneous form of equatorial jets in freely decaying shallow water turbulence. Physics of Fluids 11 (1999) 1272-1274
34. Jakob-Chien R., Hack J.J., and Williamson D.L. Spectral transform solutions to the shallow water test set. Journal of Computational Physics 119 (1995) 164-287
35. Krivodonova L., Xin J., Remacle J.-F., Chevaugeon N., and Flaherty J.E. Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Applied Numerical Mathematics 48 (2004) 323-338
36. Kubatko E.J., Westerink J.J., and Dawson C. hp Discontinuous Galerkin methods for advection dominated problems in shallow water flows. Computational Methods in Applied Mechanics and Engineering 196 (2006) 437-451
37. LeVeque R.J. Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics (2002), Cambridge University Press
38. Lomtev I., and Karniadakis G. A discontinuous Galerkin method for the Navier-Stokes equations. International Journal for Numerical Methods in Fluids 29 (1999) 587-603
39. Lomtev I., Quillen C., and Karniadakis G. Spectral/hp methods for viscous compressible flows on unstructured 2D meshes. Journal of Computational Physics 33 (1998) 325-357
40. Marchandise E., Chevaugeon N., and Remacle J.-F. Spatial and spectral superconvergence of discontinuous Galerkin method for hyperbolic problems. Journal of Computational and Applied Mathematics 215 (2006) 484-494
41. Marchandise E., Geuzaine P., Chevaugeon N., and Remacle J. A stabilized finite element method using a discontinuous level set approach for the computation of bubble dynamics. Journal of Computational Physics 225 (2007) 949-974
42. Marchandise E., and Remacle J.-F. A stabilized finite element method using a discontinuous level set approach for solving two phase incompressible flows. Journal of Computational Physics 219 (2006) 780-800
43. Nair R.D., Thomas S.J., and Loft R.D. A discontinuous Galerkin global shallow water model. Monthly Weather Review 133 (2005) 876-888
44. Pons J., Font J., Ibanez J., Marti J., and Miralles J. General relativistic hydrodynamics with special relativistic riemann solvers. Astronomy and Astrophysics 339 (1998) 638-642
45. Rancic M., Purser R., and Mesinger F. A global shallow-water model using an expanded spherical cube: gnomonic versus conformal coordinates. Quaternary Journal of Royal Meteorology Society 122 (1996) 959-982
46. W.H. Reed, T.R. Hill, Triangular mesh method for the neutron transport equation, Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.
47. J.-F. Remacle, J.E. Flaherty, M. Shephard, Parallel algorithm oriented mesh database, in: Proceedings of the 10th International Meshing Roundtable, 2001.
48. Remacle J.-F., Flaherty J.E., and Shephard M.S. An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems. SIAM Review 45 (2003) 53-72
49. Remacle J.-F., Li X., Shephard M.S., and Flaherty J.E. Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods. International Journal for Numerical Methods in Engineering 62 7 (2005) 899-923
50. Remacle J.-F., Soares Frazão S., Li X., and Shephard M. An adaptative discontinuous Galerkin method for the shallow water equations. International Journal for Numerical Methods in Fluids (2003) 1-20
51. Ronchi C., Iacono R., and Paolucci P.S. The cubed sphere: a new method for the solution of partial differential equations in spherical geometry. Journal of Computational Physics 124 (1996) 93-114
52. Rossmanith J.A. A wave propagation method for hyperbolic systems on the sphere. Journal of Computational Physics 213 (2006) 629-658
53. Rossmanith J.A., Bale D.S., and LeVeque R.J. A wave propagation algorithm for hyperbolic systems on curved manifolds. Journal of Computational Physics 199 (2004) 631-662
54. Smith R.K., and Dritschel D.G. Revisiting the Rossby-Haurwitz wave test case with contour advection. Journal of Computational Physics 217 (2006) 473-484
55. Stuhne G.R., and Peltier W.R. New icosahedral grid-point discretizations of the shallow water equations on the sphere. Journal of Computational Physics 148 (1999) 23-58
56. Swarztrauber P.N., Williamson D.L., and Drake J.B. The Cartesian method for solving partial differential equations in spherical geometry. Dynamics of Atmospheres and Oceans 27 (1997) 679-706
57. Taylor M., Tribbia J., and Iskandarani M. The spectral element method for the shallow water equations on the sphere. Journal of Computational Physics 130 (1997) 92-108
58. Thuburn J., and Li Y. Numerical simulations of Rossby-Haurwitz waves. Tellus 52A (2000) 181-189
59. Toro E. Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction (1997), Springer, Berlin
60. Warburton T., and Karniadakis G.E. A discontinuous Galerkin method for the viscous MHD equations. Journal of Computational Physics 152 (1999) 608-641
61. T. Warburton, I. Lomtev, R. Kirby, G. Karniadakis, A discontinuous Galerkin method for the Navier-Stokes equations on hybrid grids, Technical Report Number 97-14, Center for Fluid Mechanics, Division of Applied Mathematics, Brown University, 1997.
62. Williamson D.L., Drake J.B., Hack J.J., Jakob R., and Swarztrauber P.N. A standard test set for numerical approximations to the shallow water equations in spherical geometry. Journal of Computational Physics 102 (1992) 211-224