Dam-Break and Transcritical Flow Simulation of 1D Shallow Water Equations with Discontinuous Galerkin Finite Element Method

- Journal title : Journal of The Korean Society of Civil Engineers
- Volume 34, Issue 5, 2014, pp.1383-1393
- Publisher : Korean Society of Civil Engeneers
- DOI : 10.12652/Ksce.2014.34.5.1383

Title & Authors

Dam-Break and Transcritical Flow Simulation of 1D Shallow Water Equations with Discontinuous Galerkin Finite Element Method

Yun, Kwang Hee; Lee, Haegyun; Lee, Namjoo;

Yun, Kwang Hee; Lee, Haegyun; Lee, Namjoo;

Abstract

Recently, with rapid improvement in computer hardware and theoretical development in the field of computational fluid dynamics, high-order accurate schemes also have been applied in the realm of computational hydraulics. In this study, numerical solutions of 1D shallow water equations are presented with TVD Runge-Kutta discontinuous Galerkin (RKDG) finite element method. The transcritical flows such as dam-break flows due to instant dam failure and transcritical flow with bottom elevation change were studied. As a formulation of approximate Riemann solver, the local Lax-Friedrichs (LLF), Roe, HLL flux schemes were employed and MUSCL slope limiter was used to eliminate unnecessary numerical oscillations. The developed model was applied to 1D dam break and transcritical flow. The results were compared to the exact solutions and experimental data.

Keywords

Discontinuous galerkin finite element method;Approximate riemannn solver;Slope limiter;Shallow water equation;Dam-break flow;Transcritical flow;

Language

Korean

References

1.

Chavent, G. and Salzano, G. (1982). "A finite element method for the 1D water flooding problem with gravity." J. Comp. Phys., Vol. 45, pp. 307-344.

2.

Cockburn, B. (1999). "Discontinuous galerkin methods for convection dominated problems." Lecture Notes in Computational Science and Engineering, Springer, Vol. 9, pp. 69-224.

3.

Cunge, J. A., Holly, F. M. and Verwey, A. (1980). Practical aspects of computational river hydraulics, Pitman, London.

4.

Harten, A., Lax. P. D., van Leer, B. (1983). "On upstream differencing and Godunov-type schemes for hyperbolic conservation laws." SIAM Rev. Vol. 25, No. 1, pp. 35-61.

5.

Hesthaven, J. S. and Warburton, T. (2007). Nodal discontinuous galerkin methods: Algorithms, Analysis, and Applications, Springer, New York.

6.

Hughes, T. J. R. and Brooks, A. N. (1982). "A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions:Application to the Streamline-Upwind Procedure." Finite Elements in Fluids, R. H. Gallagher et al., eds., Wiley, Chichester, U.K., Vol. 4, pp. 46-65.

7.

Jain, S. C. (2001). Open-channel flow, John Wiley & Sons, New York.

8.

Kim, J. S. and Han, K. Y. (2009). "One-dimensional hydraulic modeling of open channel flow using the Riemann approximate solver - Application for natural river." J. Korea Water Resources Association, Vol. 42, No. 4, pp. 271-279 (in Korean).

9.

Lai, W. and Khan, A. A. (2012). "Discontinuous galerkin method for 1D shallow water flow in nonrectangular and nonprismatic channels." J. Hydraul. Engrg., Vol. 138, No. 3, pp. 285-296.

10.

Lax, P. D. (1954). "Weak solutions of nonlinear hyperbolic equations and their numerical computation." Comm. Pure Appl. Math., Vol. 7, pp. 159-193.

11.

Lee, H. and Lee, N. J. (2013). "Simulation of shallow water flow by discontinuous galerkin finite element method." Proc. of 2013 IAHR World Congress, Chengdu, China.

12.

LeVeque, R. J. (2002). Finite volume methods for hyperbolic problems, Cambridge University Press, Cambridge.

13.

Meselhe, E. A., Sotiropoulos, F. and Holly, F. M. (1997). "Numerical simulation of transcritical flow in open channels." J. Hydraul. Engrg., Vol. 23, No. 9, pp. 774-782.

14.

Mohapatra, P. K. and Bhallamudi, S. M. (1996). "Computation of a dam-break flood wave in channel transitions." Adv. Water Resour., Vol. 19, No. 3, pp. 181-187.

15.

Reed, W. H. and Hill, T. R. (1973). Triangular mesh methods for the neutron transport equation, Scientific Laboratory Report, Los Alamos, LA-UR-73-479.

16.

Roe, P. (1981). "Approximate riemann solvers, parameter vectors, and difference schemes." J. Comput. Phys., Vol. 43, No. 2, pp. 357-372.

17.

Schwanenberg, D. and Harms, M. (2004). "Discontinuous galerkin finite-element method for transcritical two-dimensional shallow water flows." J. Hydraul. Engrg., Vol. 130, No. 5, pp. 412-421.