Well-balanced RKDG2 solutions to the shallow water equations over irregular domains with wetting and drying

Computers and Fluids

Volumn 39, Issue 10, 2010, Pages 2040-2050

  Kesserwani, Georges ^a   Liang, Qiuhua ^a  

^a NEWCASTLE UNIVERSITY   (United Kingdom)

Author keywords

Discontinuous galerkin; Friction effects; Godunov type method; Shallow water equations; Well balanced scheme; Wetting and drying

Indexed keywords

DISCONTINUOUS GALERKIN; FRICTION EFFECTS; SHALLOW WATER EQUATIONS; TYPE METHODS; WELL-BALANCED SCHEME; WETTING AND DRYING;

CURING; FLOW OF WATER; FRICTION; GALERKIN METHODS; PIECEWISE LINEAR TECHNIQUES; RUNGE KUTTA METHODS; TRIBOLOGY; WATER LEVELS; WETTING;

DRYING;

EID: 77956338382     PISSN: 00457930     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.compfluid.2010.07.008     Document Type: Article

References (57)

1. 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.
2. Aureli F, Mignosa P, Tomirotti M. Dam-break flows in presence of abrupt bottom variations. In: Proc XXVIII IAHR Congr, Graz, Australia; 1999. p. 163-71.
3. Begnudelli L., Sanders B.F. Conservative wetting and drying methodology for quadrilateral grid finite-volume models. J Hydrol Eng 2007, 133(3):312-322.
4. Berthon C., Marche F. A positive preserving high order VFROE scheme for shallow water equations: a class of relaxation schemes. SIAM J Sci Comput 2008, 30(5):2587-2612.
5. Bokhove O. Flooding and drying in discontinuous Galerkin finite element method for shallow water flows. J Sci Comput 2005, 22-23(1):47-82.
6. Brufau P., García-Navarro P., Vázquez-Cendón M.E. A numerical model for the flooding and drying of irregular domains. Int J Numer Methods Fluids 2002, 39(3):247-275.
7. Brufau P., García-Navarro P., Vázquez-Cendón M.E. Zero mass error using unsteady wetting-drying conditions in shallow flows over dry irregular topography. Int J Numer Methods Fluids 2004, 45(10):1047-1082.
8. 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.
9. Burguete J., García-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 Methods Fluids 2008, 58(4):403-425.
10. Casulli V. A high-resolution wetting and drying algorithm for free-surface hydrodynamics. Int J Numer Methods Fluids 2009, 60(4):391-408.
11. Cockburn B., Lin S., Shu C.W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J Comput Phys 1989, 84(1):90-113.
12. Cockburn B., Shu C.W. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J Sci Comput 2001, 16(3):173-261.
13. 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.
14. Cockburn B. Discontinuous Galerkin methods. ZAMM 2003, 83(11):731-754.
15. Ern A., Piperno S., Djadel K. A well-balanced Runge-Kutta discontinuous Galerkin method for the shallow-water equations with ooding and drying. Int J Numer Methods Fluids 2008, 58(1):1-25.
16. 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.
17. Gallouët T., Hérard J.M., Seguin N. Some approximate Godunov schemes to compute shallow-water equations with topography. Comput Fluids 2003, 32(4):479-513.
18. Guinot V. Godunov-type schemes: an introduction for engineers 2003, Elsevier, Amsterdam.
19. Heniche M., Secretan Y., Boudreau P., Leclerc M. A two-dimensional finite element drying-wetting shallow water model for rivers and estuaries. Adv Water Resour 2000, 23(4):359-372.
20. Kesserwani G., Ghostine R., Vazquez J., Ghenaim A., Mosé R. Application of a second order Runge-Kutta discontinuous Galerkin scheme for the shallow water equations with source terms. Int J Numer Methods Fluids 2008, 56(7):805-821.
21. Gourgue O., Comblen R., Lambrechts J., Kärnä T., Legat V., Deleersnijder E. A flux-limiting wetting-drying method for finite-element shallow-water models, with application to the Scheldt Estuary. Adv Water Resour 2009, 32(12):1726-1739.
22. Kesserwani G., Liang Q., Vazquez J., Mosé R. Well-balancing issues related to the RKDG2 scheme for the shallow water equations. Int J Numer Methods Fluids 2010, 62(4):428-448.
23. Nicolos I.K., Delis A.I. An unstructured node-centred finite volume scheme for shallow water flows with wet/dry fronts over complex topography. Comput Methods Appl Mech Eng 2009, 198:3723-3750.
24. Liang Q., Borthwick A.G.L. Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography. Comput Fluids 2009, 38(2):221-234.
25. Marche F., Bonneton P., Fabrie P., Seguin N. Evaluation of well-balanced bore-capturing schemes for 2D wetting and drying processes. Int J Numer Methods Fluids 2007, 53(5):867-894.
26. Noelle S., Pankratz N., Puppo G., Natvig J.R. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J Comput Phys 2006, 213(2):474-499.
27. Rogers B.D., Borthwick A.G.L., Taylor P.H. Mathematical balancing of flux gradient and source terms prior to using Roe's approximate Riemann solver. J Comput Phys 2003, 192(2):422-451.
28. Schwanenberg D. Die Runge-Kutta-Discontinuous-Galerkin-Methode zur Lösung konvektionsdominierter tiefengemittelter Flachwasserprobleme. PhD thesis, Fakultät für Bauingenieurwesen; 2003.
29. Shu C.W., Osher S. Efficient implementation of essentially non-oscillatory shock capturing schemes. J Comput Phys 1988, 77(2):439-471.
30. Toro E.F., García-Navarro P. Godunov-type methods for free-surface shallow flows: a review. J Hydraul Res 2007, 45(6):737-751.
31. Toro E.F. Shock-capturing methods for free-surface shallow flows 2001, John Wiley & Sons, Ltd.
32. 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.
33. Zhao D.H., Shen H.W., Tabios G.Q., Lai J.S., Tan W.Y. Finite volume two-dimensional unsteady flow model for river basins. J Hydraul Eng 1994, 120(7):863-883.
34. 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.
35. Liang Q., Marche F. Numerical resolution of well-balanced shallow water equations with complex source terms. Adv Water Resour 2009, 32:873-884.
36. Yoon T.H., Kang S.-K. Finite volume model for two-dimensional shallow water flows on unstructured grids. ASCE J Hydraul Eng 2004, 130(7):678-688.
37. Sweby P.K. High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J Numer Anal 1984, 21(5):995-1011.
38. Thacker W.C. Some exact solutions to the nonlinear shallow-water wave equations. J Fluid Mech 1981, 107:499-508.
39. Hiver JM. Adverse-slope and slope (bump). In: Soares Frazão S, Morris M, Zech Y, editors. Concerted action on dam break modelling: objectives, project report, test cases, meeting proceedings. Université catholique de Louvain, Civ. Eng. Dept., Hydraulics Division, Louvain-la-Neuve, Belgium, CD-ROM; 2000.
40. Murillo J., García-Navarro P., Burguete J. Time step restrictions for well-balanced shallow water solutions in non-zero velocity steady states. Int J Numer Methods Fluids 2009, 60(12):1351-1377.
41. Van Leer B. Upwind and high-resolution methods for compressible flow: from donor cell to residual-distribution schemes. Commun Comput Phys 2006, 1(2):192-206.
42. Lee S.H., Wright N.G. Simple and efficient solution of the shallow water equations with source terms. Int J Numer Methods Fluids 2010, 63(3):313-340.
43. 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 Methods Fluids 2009, 59(5):535-557.
44. 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.
45. Nair R.D., Thomas S.J., Loft R.D. A discontinuous Galerkin global shallow water model. Mon Weather Rev 2005, 133(4):876-888.
46. Eskilsson C., Sherwin S.J. A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations. Int J Numer Methods Fluids 2004, 45(6):605-623.
47. Kubatko E.J., Westerink J.J., Dawson C. hp Discontinuous Galerkin methods for advection dominated problems in shallow water flow. Comput Methods Appl Mech Eng 2006, 96(1-3):437-451.
48. Aizinger V., Dawson C. A discontinuous Galerkin method for two-dimensional flow and transport in shallow water. Adv Water Resour 2002, 25(1):67-84.
49. Ambati V.R., Bokhove O. Space-time discontinuous Galerkin discretization of rotating shallow water equations. J Comput Phys 2007, 225(2):1233-1261.
50. Van Leer B. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method. J Comput Phys 1979, 32(1):101-136.
51. Sampson J., Easton A., Singh M. Moving boundary shallow water flow above parabolic bottom topography. ANZIAM J 2006, 47(EMAC2005):C373-C387.
52. Burguete J., García-Navarro P., Murillo J., Garcia-Palacin I. Analysis of the friction term in one-dimensional shallow water model: application to open channel and river flow. J Hydraul Eng 2007, 133(9):1048-1063.
53. Liang Q. Flood simulation using a well-balanced shallow flow model. ASCE J Hydraul Eng 2010, 10.1061/(ASCE)HY.1943-7900.000021.
54. 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.
55. 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.
56. 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.
57. 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.