Discontinuous galerkin method for 1D shallow water flows in natural rivers

Engineering Applications of Computational Fluid Mechanics

Volumn 6, Issue 1, 2012, Pages 74-86

  Lai, W ^a   Khan, A A ^a  

^a South   (United States)

Author keywords

Natural rivers; Runge Kutta discontinuous Galerkin; Saint Venant equations; Shallow water flows; Total variation diminishing

Indexed keywords

EID: 84858658007     PISSN: 19942060     EISSN: 1997003X     Source Type: Journal    
DOI: 10.1080/19942060.2012.11015404     Document Type: Article

References (43)

1. Bokhove O (2005). Flooding and drying in discontinuous Galerkin finite-element discretizations of shallow-water equations. Part 1: One dimension. Journal of Scientific Computing 22-23(1-3):47-82. (Pubitemid 40537847)
2. Capart H, Eldho TI, Huang SY, Young DL, Zech Y (2003). Treatment of natural geometry in finite volume river flow computations. Journal of Hydraulic Engineering 125(5):385-393. (Pubitemid 36624114)
3. Catella M, Paris E, Solari L (2008). Conservative scheme for numerical modeling of flow in natural geometry. Journal of Hydraulic Engineering 134(6):736-748. (Pubitemid 351712296)
4. Cockburn B (2001). Discontinuous Galerkin methods for convection dominated problems. Lecture Notes in Computational Science and Engineering, Springer, Berlin, Vol. 9, 69-224.
5. Cockburn B, Hou S, Shu CW (1990). The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case. Mathematics of Computation 54(190):545-581.
6. Cockburn B, Lin SY, Shu CW (1989). TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems. Journal of Computational Physics 84(1):90-113.
7. Cockburn B, Shu CW (1989). TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework. Mathematics of Computation 52(186):411-435.
8. Cockburn B, Shu CW (1998). The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems. Journal of Computational Physics 141(2):199-224. (Pubitemid 128344796)
9. Cunge JA, Holly FM, Verwey A (1980). Practical Aspects of Computational River Hydraulics. Pitman, London.
10. Einfeldt B (1988). On Godunov-type methods for gas dynamics. SIAM Journal on Numerical Analysis 25(2):294-318.
11. Emmett WM, Myrick RM, Meade RH (1980). Field data describing the movement and storage of sediment in the East Fork River, Wyoming, Part I. River hydraulics and sediment transport, 1979. USGS Open-File Report 80-1189, Denver.
12. Ern A, Piperno S, Djadel K (2008). A well-balanced Runge-Kutta discontinuous Galerkin method for the shallow-water equations with flooding and drying. International Journal for Numerical Methods in Fluids 58(1):1-25.
13. Erpicum S, Dewals B, Archambeau P, Detrembleur S, Pirotton M (2010). Detailed inundation modeling using high resolution DEMs. Engineering Applications of Computational Fluid Mechanics 4(2):196-208.
14. Fraccarollo L, Toro EF (1995). Experimental and numerical assessment of the shallow water model for two-dimensional dam-break type problems. Journal of Hydraulic Research 33(6):843-846.
15. Frãzao SS, Testa G (1999). The Toce River test case: Numerical results analysis. Proceedings of the 3rd CADAM Workshop, Milan, Italy.
16. Garcia-Navarro P, Vázquez-Cendón M (2000). On numerical treatment of the source terms in the shallow water equations. Computers & Fluids 29(8):951-979.
17. Gharangik AM, Chaudhry MH (1991). Numerical simulation of hydraulic jump." Journal of Hydraulic Engineering 117(9):1195-1211.
18. Godunov SK (1959). A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Math Sbomik 47(3):271-306.
19. Gottlieb S, Shu CW (1998). Total variation diminishing Runge-Kutta schemes. Mathematics of Computation 67(221):73-85. (Pubitemid 128381509)
20. Harten A, Lax PD, Van Leer B (1983). On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Review 25(1):35-61.
21. Henderson FM (1996). Open Channel Flow. McGraw-Hill, New York.
22. Kesserwani G, Ghostine R, Vazquez J, Ghenaim A, Mosé R (2008). Application of a second-order Runge-Kutta discontinuous Galerkin scheme for the shallow water equations with source terms. International Journal for Numerical Methods in Fluids 56(7):805-821. (Pubitemid 351314827)
23. Kesserwani G, Liang Q, Vazquez J, Mosé R (2010). Well-balancing issues related to the RKDG2 scheme for the shallow water equations. International Journal for Numerical Methods in Fluids 62(4):428-448.
24. Khalifa AM (1980). Theoretical and Experimental Study of the Radial Hydraulic Jump. Ph.D. Thesis, University of Windsor, Windsor, Ontario, Canada.
25. Krivodonova L, Xin J, Remacle J-F, Chevaugeon N, Flaherty JE (2004). Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Applied Numerical Mathematics 48(3-4):323-338.
26. Li BQ (2006). Discontinuous Finite Element in Fluid Dynamics and Heat Transfer. Springer, London.
27. Meade RH, Myrick RM, Emmett WW (1980). Field data describing the movement and storage of sediment in the East Fork River, Wyoming, Part II. Bed elevations, 1979. US Geological Survey, Open-File Report 80-1190, Denver.
28. Nujić M (1995). Efficient implementation of non-oscillatory schemes for the computation of free-surface flows. Journal of Hydraulic Research 33(1):101-111.
29. Perthame B, Simeoni C (2001). A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38(4):201-231. (Pubitemid 33743526)
30. Ray HA, Kjelstrom LC (1976). The flood in southeastern Idaho from the Teton Dam failure of June 5, 1976. U.S. Geological Survey, Open-File Report 77-765, Boise, Idaho, USA.
31. Reed WH, Hill T (1973). Triangular mesh method for the neutron transport equation. Los Alamos Report LA-UR-73-479.
32. Rhebergen S, Bokhove O, Van Der Vegt JJW (2008). Discontinuous Galerkin finite element methods for hyperbolic nonconservative partical differential equations. Journal of Computational physics 227(3):1887-1922.
33. Roe P (1981). Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics 43(2):357-372.
34. Rosu C, Ahmed M (1999). Toce River dam-break test case - A comparison between the ISIS numerical model and the physical model. Proceedings of the 3rd CADAM Workshop, Milan, Italy.
35. Schwanenberg D, Köngeter J (2000). A discontinuous Galerkin method for the shallow water equations with source terms. Lecture Notes in Computational Science and Engineering, Springer, Berlin, Vol. 11, 419-424.
36. Schoklitsch A (1917). Ueber Dammbruchwellen. Sitzungsberichte der Kaiserlichen Akademie Wissenschaften, Viennal, 126,1489-1514.
37. Toro E (1989). A weighted average flux method for hyperbolic conservation laws. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 423(1865):401-418.
38. Toro E, Spruce M, Speares W (1994). Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4(1):25-34.
39. Tritthart M, Gutknecht D (2007). Three-dimensional simulation of free-surface flows using polyhedral finite volumes. Engineering Applications of Computational Fluid Mechanics 1(1):1-14.
40. Xing Y, Zhang X, Shu C (2010). Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Advances in Water Resources 33(12):1476-1493.
41. Ying X, Jorgeson J, Wang SSY (2009), Modeling dam-break flows using finite volume method on unstructured grid. Engineering Applications of Computational Fluid Mechanics 3(2):184-194.
42. Ying X, Khan AA, Wang SSY (2004). Upwind conservative scheme for the Saint Venant equations. Journal of Hydraulic Engineering 130(10):977-987. (Pubitemid 39519760)
43. Zhou JG, Causon DM, Mingham CG, Ingram DM (2001). The surface gradient method for the treatment of source terms in the shallow-water equations. Journal of Computational Physics 168(1):1-25. (Pubitemid 33668235)