Dynamically adaptive grid based discontinuous Galerkin shallow water model

Advances in Water Resources

Volumn 37, Issue , 2012, Pages 23-39

  Kesserwani, Georges ^a   Liang, Qiuhua ^b  

^a UNIVERSITY OF SHEFFIELD   (United Kingdom)

^b NEWCASTLE UNIVERSITY   (United Kingdom)

Author keywords

Computational efficiency; Discontinuous Galerkin method; Dynamic grid adaptation; Inundation modeling; Quadrilateral mesh; Shallow water equations

Indexed keywords

ADAPTIVE GRIDS; ADAPTIVE MODELS; COMPUTATIONAL COSTS; DAM-BREAKS; DIAGNOSTIC TESTS; DISCONTINUOUS GALERKIN; DISCONTINUOUS GALERKIN METHODS; DYNAMIC GRID ADAPTATION; DYNAMIC GRID ADAPTATIONS; FLOOD MODELING; FLOW DATA; GRID ADAPTATION; GRID SYSTEMS; GRID-BASED; LABORATORY SCALE; LOCAL FLOWS; MOVING WAVES; NUMERICAL EVIDENCE; NUMERICAL MODELS; QUADRILATERAL MESH; QUADRILATERAL MESHES; RUNGE-KUTTA DISCONTINUOUS GALERKIN; SHALLOW WATER EQUATIONS; SHALLOW WATER MODEL; SHALLOW WATER WAVES;

COMPUTATIONAL EFFICIENCY; COMPUTER SIMULATION; EQUATIONS OF MOTION; FLOOD CONTROL; FLOODS; RUNGE KUTTA METHODS;

GALERKIN METHODS;

ALGORITHM; ANALYTICAL METHOD; COMPUTER SIMULATION; FLOOD; GALERKIN METHOD; NUMERICAL METHOD; NUMERICAL MODEL; SHALLOW-WATER EQUATION; WATER WAVE;

EID: 84855293875     PISSN: 03091708     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.advwatres.2011.11.006     Document Type: Article

References (79)

1. Aizinger V., Dawson C. A discontinuous Galerkin method for two-dimensional flow and transport in shallow water. Adv Wat Res 2002, 25(1):67-84.
2. Ambati V.R., Bokhove O. Space-time discontinuous Galerkin discretization of rotating shallow water equations. J Comput Phys 2007, 225(2):1233-1261.
3. Audusse E., Bouchut F., Bristeau M.O., Klein R., Perthame B. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J Sci Comput 2004, 25(6):2050-2065.
4. Aureli F., Maranzoni A., Mignosa P., Ziveri C. A weighted surface-depth gradient method for the numerical integration of the 2D shallow water equations with topography. Adv Wat Res 2008, 31(7):962-974.
5. Bader M., Bock C., Schwaiger J., Scaba B. Dynamically adaptive simulations with minimal memory requirement-solving the shallow water equation using sierpinski curves. Siam J Sci Comput 2010, 32(1):212-228.
6. Begnudelli L., Sanders B.F., Bradford S.F. Adaptive Godunov-Based Model for Flood Simulation. J Hydr Eng 2008, 134(6):714-725.
7. Begnudelli L., Sanders B.F. Conservative wetting and drying methodology for quadrilateral grid finite-volume models. J Hydr Eng 2007, 133(3):312-322.
8. Begnudelli L., Sanders B.F. Simulation of the St. Francis dam-break flood. J Eng Mech 2007, 133(11):1200-1212.
9. Benkhaldoun F., Elmahi I., Seaid M. A new finite volume method for flux-gradient and sour-term balancing in shallow water equations. Comput Meth App Mech Eng 2010, 199:3324-3335.
10. Bernard P.E., Remacle J.F., Legat V. Boundary discretization for high order discontinuous Galerkin computations of tidal flows around shallow water islands. Int J Numer Meth Fluids 2009, 59(5):535-557.
11. Bokhove O. Flooding and drying in discontinuous Galerkin finite element method for shallow water flows. J Sci Comput 2005, 22-23(1):47-82.
12. Bradford S.F., Sanders B.F. Finite volume model for shallow water flooding of arbitrary topography. J Hydr Eng 2002, 128(3):289-298.
13. Brufau P., Garci{dotless}́a-Navarro P., Vázquez-Cendón M.E. Zero mass error using unsteady wetting-drying conditions in shallow flows over dry irregular topography. Int J Numer Meth Fluids 2004, 45(10):1047-1082.
14. Bunya S., Kubatko E.J., Westerink J.J., Dawson C. A wetting and drying treatment for the Runge-Kutta discontinuous Galerkin solution to the shallow water equations. Comput Meth Appl Mech Eng 2009, 198(17-20):1548-1562.
15. Burguete J., Garci{dotless}́a-Navarro P., Murillo J., Garci{dotless}́a-Palaci{dotless}́n I. Analysis of the friction term in the one-dimensional shallow water model. J Hydr Eng 2007, 133(9):1048-1063.
16. Burguete J., Garci{dotless}́a-Navarro P., Murillo J. Friction term discretization and limitation to preserve stability and conservation in the 1D shallow-water model: application to unsteady irrigation and river flow. Int J Numer Meth Fluids 2008, 58(4):403-425.
17. Caleffi V., Valiani A., Bernini A. Fourth-order balanced source term treatment in central WENO schemes for shallow water equations. J Comput Phys 2007, 218:228-245.
18. Casulli V. A high-resolution wetting and drying algorithm for free-surface hydrodynamics. Int J Numer Meth Fluids 2009, 60(4):391-408.
19. Cockburn B., Shu C.W. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J Sci Comput 2001, 16(3):173-261.
20. Cockburn B., Shu C.W. The Runge-Kutta local projection P1-discontinuous Galerkin method for scalar conservation laws. RAIRO Modél Math Anal Numér 1991, 25(3):337-361.
21. Ern A., Piperno S., Djadel K. A well-balanced Runge-Kutta discontinuous Galerkin method for the shallow-water equations with flooding and drying. Int J Numer Meth Fluids 2008, 58(1):1-25.
22. Eskilsson C., Sherwin S.J. A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations. Int J Numer Meth Fluids 2004, 45(6):605-623.
23. Fagherazzi S., Rasetarinera P., Hussaini Y.M., Furbish D.J. Numerical solution of the dam-break problem with a discontinuous Galerkin method. J Hydr Eng 2004, 130(6):532-539.
24. Gallardo J.M., Parés C., Castro M. On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas. J Comput Phys 2007, 227(1):574-601.
25. Gallegos H.A., Schubert J.E., Sanders B.F. Two-dimensional, high-resolution modeling of urban dam-break flooding: A case study of Baldwin Hills, California. Adv Wat Res 2009, 32(8):1323-1335.
26. George D.L., LeVeque R. Finite volume methods and adaptive refinement for global tsunami propagation and local inundation. Sci Tsunami Hazards 2006, 24(5):319-328.
27. George DL Adaptive finite volume methods with well-balanced Riemann solvers for modeling floods in rugged terrain: Application to the Malpasset dam-break flood (France, 1959). Int J Numer Meth Fluids 2011, 10.1002/fld.2298.
28. Giraldo F.X., Restelli M. High-order semi-implicit time-integrators for a triangular discontinuous Galerkin oceanic shallow water model. Int J Numer Meth Fluids 2010, 63(9):1077-1102.
29. Giraldo F.X., Warburton T. A high-order triangular discontinuous Galerkin oceanic shallow water model. Int J Numer Meth Fluids 2008, 56(7):899-925.
30. Giraldo F.X. High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere. J Comput Phys 2006, 214(2):447-465.
31. Gourgue O., Comblen R., Lambrechts J., Kärnä T., Legat V., Deleersnijder E. A flux-limiting wetting-drying for finite element shallow-water models, with application to Scheldt Estuary. Adv Wat Res 2009, 32(12):1726-1739.
32. Greaves D.M., Borthwick A.G.L. Hierarchical tree-based finite element mesh generation. Int J Numer Meth Eng 1999, 45:447-471.
33. Guinot V. Godunov-type schemes: an introduction for engineers 2003, Elsevier, Amsterdam.
34. Hunter N.M., Bates P.D., Neelz S., Pender G., Villanueva I., Wright N.G., et al. Benchmarking 2D hydraulic models for urban flood simulations. Proc Inst Civ Eng-Wat Mgmt 2008, 161(1):13-30.
35. Kärnä T., de Brye B., Gourgue O., Lambrechts J., Comblen R., Legat V., et al. A fully implicit wetting-drying method for DG-FEM shallow water models, with an application to the Scheldt Estuary. Comput Methods Appl Mech Eng 2011, 200(5-8):509-524.
36. Kesserwani G., Liang Q., Vazquez J., Mosé R. Well-balancing issues related to the RKDG2 scheme for the shallow water equations. Int J Numer Meth Fluids 2010, 62(4):428-448.
37. Kesserwani G., Liang Q. A discontinuous Galerkin algorithm for the two-dimensional shallow water equations. Comput Meth App Meth Eng 2010, 199(49-52):3356-3368.
38. Kesserwani G, Liang Q Locally limited and fully conserved RKDG2 shallow water solutions with wetting and drying. J Sci Comput 2011, 10.1007/s10915-011-9476-4.
39. Krámer T., Józsa J. Solution-adaptivity in modelling complex shallow flows. Comput Fluids 2007, 36(3):562-577.
40. Krivodonova L., Xin J., Remacle J.F., Chevaugeon N., Flaherty J.E. Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl Numer Math 2004, 48(3-4):323-338.
41. Krivodonova L. Limiters for high-order discontinuous Galerkin methods. J Comput Phys 2007, 226(1):879-896.
42. Kubatko E.J., Bunya S., Dawson C., Westerink J.J., Mirabito C. A performance comparison of continuous and discontinuous finite element shallow water models. J Sci Comput 2009, 40(1-3):315-339.
43. Kubatko E.J., Bunya S., Dawson C., Westerink J.J. Dynamic p-adaptive Runge-Kutta discontinuous Galerkin methods for the shallow water equations. Comput Meth App Mech Eng 2009, 198(21-26):1766-1774.
44. Kubatko E.J., Westerink J.J., Dawson C. Hp Discontinuous Galerkin methods for advection dominated problems in shallow water flow. Comput Meth App Meth Eng 2006, 96(1-3):437-451.
45. Xu Y., Shu C.W. Local discontinuous Galerkin methods for high-order time dependent partial differentiable equations. Commun Comput Phys 2010, 7(1):1-46.
46. Lamby P., Müller S., Stiriba Y. Solution of shallow water equations using fully adaptive multiscale schemes. Int J Numer Meth Fluids 2005, 49:417-437.
47. Lee S.H., Wright N.G. A simple and efficient solution of the shallow water equations with source terms. Int J Numer Meth Fluids 2010, 63(3):313-340.
48. LeVeque R.J. Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm. J Comput Phys 1998, 146(1):346-365.
49. Liang Q., Du G., Hall J.W., Borthwick A.G.L. Flood inundation modelling with an adaptive quadtree grid shallow water equation solver. J Hydr Eng 2008, 134(11):1603-1610.
50. Liang Q., Marche F. Numerical resolution of well-balanced shallow water equations with complex source terms. Adv Wat Res 2009, 32:873-884.
51. Liang Q A structured but non-uniform Cartesian grid-based model for the shallow water equations. Int J Numer Meth Fluids 2010, 10.1002/fld.2266.
52. Liang Q. Flood simulation using a well-balanced shallow flow model. J Hydr Eng 2010, 136(9):69-675.
53. Marche F., Bonneton P., Fabrie P., Seguin N. Evaluation of well-balanced bore-capturing schemes for 2D wetting and drying processes. Int J Numer Meth Fluids 2007, 53(5):867-894.
54. Mungkasi S., Roberts S.G. On the best quantity reconstructions for a well-balanced finite volume method used to solve the shallow water wave equations with a wet/dry interface. ANZIAM J. 51(EMAC2009) 2010, C48-C65.
55. Nair R.D., Thomas S.J., Loft R.D. A discontinuous Galerkin global shallow water model. Mon Weather Rev 2005, 133(4):876-888.
56. Nikolos I.K., Delis A.I. An unstructured node-centred finite volume scheme for shallow water flows with wet/dry fronts over complex topography. Comput Meth App Mech Eng 2010, 198:3723-3750.
57. Remacle J.F., Soares-Frazão S., Li X., Shephard M.S. An adaptive discretization of the shallow water equations based on discontinuous Galerkin methods. Int J Numer Meth Fluids 2006, 52(8):903-923.
58. Schwanenberg D., Harms M. Discontinuous Galerkin finite-element method for tanscritical two-dimensional shallow water flow. J Hydr Eng 2004, 130(5):412-421.
59. Schwanenberg D. Die Runge-Kutta-Discontinuous-Galerkin-Methode zur Lösung konvektionsdominierter tiefengemittelter Flachwasserprobleme. Ph.D. thesis; Fakultät für Bauingenieurwesen, 2003.
60. Skoula Z.D., Borthwick A.G.L., Moutzouris C.I. Godunov-type solution of the shallow water equations on adaptive unstructured triangular grids. Int J Comput Fluid Dynam 2006, 20(9):621-636.
61. Tang H. Solution of the shallow-water equations using an adaptive moving mesh method. Int J Numer Meth Fluids 2004, 44:789-810.
62. Tassi P.A., Bokhove O., Vionnet C.A. Space discontinuous Galerkin method for shallow water flows-kinetic and HLLC flux, and potential vorticity generation. Adv Wat Res 2007, 30(4):998-1015.
63. Toro E.F., Garci{dotless}́a-Navarro P. Godunov-type methods for free-surface shallow flows: A review. J Hydr Res 2007, 45(6):737-751.
64. Toro E.F. Shock-capturing methods for free-surface shallow flows 2001, John Wiley & Sons, Ltd.
65. Valiani A., Caleffi V., Zanni A. Case Study: Malpasset dam-break simulation using a two-dimensional finite volume method. J Hydr Eng 2002, 128(5):460-472.
66. Wang Y, Liang Q, Kesserwani G, Hall WJ A positivity-preserving zero-inertia model for flood simulation. Comput Fluids 2011, 10.1016/j.compfluid.2011.01.026.
67. Wirasaet D, Tanaka S, Kubatko EJ, Westerink JJ, Dawson C A performance comparison of nodal discontinuous Galerkin methods on triangles and quadrilaterals. Int J Numer Meth Fluids 2011, 10.1002/fld.2376.
68. Xing Y., Shu C.W. A new approach of high order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. Commun Comput Phys 2006, 1(1):101-135.
69. Xing Y., Shu C.W. High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. J Comput Phys 2006, 214(2):567-598.
70. Xing Y., Zhang X., Shu C.W. Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv Wat Res 2010, 33(12):, 1476-1493.
71. Yoon T.H., Kang S.K. Finite volume model for two-dimensional shallow water flows on unstructured grids. ASCE J Hydr Eng 2004, 130(7):678-688.
72. Zhang M., Shu C.W. An analysis of and a comparison between the discontinuous Galerkin and the spectral finite volume methods. Comput Fluids 2003, 34(4-5):581-592.
73. Zhou J.G., Causon D.M., Ingram D.M., Mingham C.G. The surface gradient method for the treatment of source terms in the shallow-water equations. J Comput Phys 2001, 168(1):1-25.
74. Zhou T., Li Y., Shu C.W. Numerical comparison of WENO finite volume and Runge-Kutta discontinuous Galerkin methods. J Sci Comput 2001, 16(2):145-171.
75. Zokagoa J.M., Soulaimani A. Modeling of wetting-drying transitions in free surface flows over complex topographies. Comput Methods Appl Mech Eng 2010, 199(33-36):, 2281-2304.
76. Liang Q A simplified adaptive cartesian grid system for solving the 2D shallow water equations. Int J Numer Meth Fluids 2011, 10.1002/fld.2568.
77. Thacker W.C. Some exact solutions to the nonlinear shallow water wave equations. J Fluid Mech 1981, 107:499-508.
78. Delis A.I., Nikolos I.K., Kazolea M. Performance and Comparison of Cell-Centered and Node-Centered Unstructured Finite Volume Discretizations for Shallow Water Free Surface Flows. Arch Comput Methods Eng 2011, 18:57-118.
79. Kesserwani G, Liang Q Influence of TVD slope-limiting on local discontinuous Galerkin solutions of the shallow water equations. ASCE J Hydr Eng 2011, 10.1061/(ASCE)HY.1943-7900.0000494.