Riemann Solvers with Runge-Kutta Discontinuous Galerkin Schemes for the 1D Shallow Water Equations

Journal of Hydraulic Engineering

Volumn 134, Issue 2, 2008, Pages 243-255

  Kesserwani, G ^a   Ghostine, R ^a   Vazquez, J ^a   Ghenaim, A ^b   Mose R ^a  

^a UNIVERSITÉ DE STRASBOURG   (France)

^b INSA DE STRASBOURG   (France)

Author keywords

Finite element method; Methodology; Numerical analysis; Shallow water

Indexed keywords

EQUATIONS OF MOTION; GALERKIN METHODS; NUMERICAL METHODS; RUNGE KUTTA METHODS; COMPUTATIONAL FLUID DYNAMICS; FINITE ELEMENT METHOD;

1D SHALLOW WATER EQUATIONS; DISCONTINUOUS GALERKIN SCHEMES; HIGH-ORDER; HIGHER-ORDER; METHODOLOGY; RIEMANN SOLVER; RUNGE-KUTTA DISCONTINUOUS GALERKIN; RUNGE-KUTTA DISCONTINUOUS GALERKIN METHODS; SHALLOW WATERS; SPECTRA'S;

FINITE ELEMENT METHOD; HYDRAULICS;

LAX-FRIEDRICH SOLVER; RIEMANN SOLVER; RUNGE-KUTTA DISCONTINUOUS GALERKIN SCHEME; SHALLOW WATER EQUATION;

COMPUTATIONAL FLUID DYNAMICS; FINITE ELEMENT METHOD; GALERKIN METHODS; HYDRAULICS; RUNGE KUTTA METHODS;

ERROR ANALYSIS; FINITE ELEMENT METHOD; GALERKIN METHOD; HYDRAULICS; ONE-DIMENSIONAL MODELING; PERFORMANCE ASSESSMENT; SHALLOW-WATER EQUATION;

EID: 38349119964     PISSN: 07339429     EISSN: 19437900     Source Type: Journal    
DOI: 10.1061/(ASCE)0733-9429(2008)134:2(243)     Document Type: Article

References (35)

1. Alcrudo, F., and, Benkhaldoun, F. (2001). " Exact solutions to the Riemann problem of the shallow water equations with a bottom step." Comput. Fluids CPFLBI 0045-7930, 30 (6), 643-671. 10.1016/S0045-7930(01)00013-5
2. Babuska, I. (1988). " The p-and hp-versions of the finite element method. The state of the art." Finite elements: Theory and applications, Springer, New York.
3. Biswas, R., Devine, K. D., and, Flaherty, J. (1994). " Parallel, adaptive finite element methods for conservation laws." Appl. Numer. Math. ANMAEL 0168-9274, 14 (1-3), 255-283. 10.1016/0168-9274(94)90029-9
4. Cockburn, B. (2003). " Discontinuous Galerkin methods." ZAMM ZAMMAX 0044-2267, 83 (11), 731-754. 10.1002/zamm.200310088
5. Cockburn, B., Hou, S., and, Shu, C.-W. (1990). " The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case." Math. Comput. MCMPAF 0025-5718, 54 (190), 545-581. 10.2307/2008501
6. Cockburn, B., Karniadakis, G. E., and, Shu, C.-W. (2000). " The development of discontinuous Galerkin methods." Discontinuous Galerkin methods. Theory, computation and applications, Lecture Notes in Computational Science and Engineering, Vol. 11, B. Cockburn, G. E. Karniadakis, and C.-W. Shu., eds., Springer, New York, 3-50.
7. Cockburn, B., Lin, S.-Y., and, Shu, C.-W. (1989). " TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems." J. Comput. Phys. JCTPAH 0021-9991, 84 (1), 90-113. 10.1016/0021-9991(89)90183-6
8. Cockburn, B., and, Shu, C.-W. (1989). " TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework." Math. Comput. MCMPAF 0025-5718, 52 (186), 411-435. 10.2307/2008474
9. Cockburn, B., and, Shu, C.-W. (1991). " The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws." Math. Modell. Numer. Anal. MMNAAB 0764-583X, 25, 337-361.
10. Cockburn, B., and, Shu, C.-W. (2001). " Runge-Kutta discontinuous Galerkin method for convection-dominated problems." J. Sci. Comput. JSCOEB 0885-7474, 16 (3), 173-261. 10.1023/A:1012873910884
11. Cockburn, B., and, Shu, C.-W. (2005). " Foreword (for the special issue on discontinuous Galerkin methods)." J. Sci. Comput. JSCOEB 0885-7474, 22-23, 1-2.
12. Davis, S. F. (1988). " Simplified second-order Godunov-type methods." SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. SIJCD4 0196-5204, 9 (3), 445-473.
13. Delis, A. I. (2003). " Improved application of the HLLE Riemann solver for the shallow water equations with source terms." Commun. Numer. Methods Eng. CANMER 1069-8299, 19 (1), 59-83. 10.1002/cnm.570
14. Delis, A. I., Skeels, C. P., and, Ryrie, S. C. (2000). " Evaluation of some approximate Riemann solvers for transient open channel flow." J. Hydraul. Res. JHYRAF 0022-1686, 38 (3), 217-231.
15. Einfeldt, B. (1988). " On Godunov-type methods for gas dynamics." SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. SJNAAM 0036-1429, 25 (2), 294-318. 10.1137/0725021
16. Engquist, B., and, Osher, S. (1981). " One sided difference approximation for nonlinear conservation laws." Math. Comput. MCMPAF 0025-5718, 36 (154), 321-351. 10.2307/2007646
17. Godunov, S. K. (1959). " Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics." Math. USSR. Sb. ZZZZZZ 0025-5734, 47 (3), 271-306.
18. Harten, A., Lax, P. D., and, van Leer, B. (1983). " On upstream differencing and Godunov-type schemes for hyperbolic conservation laws." SIAM Rev. SIREAD 0036-1445, 25 (1), 35-61. 10.1137/1025002
19. Hirsch, C. (1990). " Numerical computation of internal and external flows." Computational methods for inviscid and viscous flows, Vol. 2, Wiley, Chichester, U.K.
20. Hubbard, M. E., and, Garcia-Navarro, P. (2000). " Flux difference splitting and the balancing of source terms and flux gradients." J. Comput. Phys. JCTPAH 0021-9991, 165 (1), 89-125. 10.1006/jcph.2000.6603
21. Kesserwani, G., Ghostine, R., Vazquez, J., Ghenaim, A., and, Mosé, R. (2007). " Application of a second order Runge-Kutta discontinuous Galerkin scheme for the shallow water equations with source terms." Int. J. Numer. Methods Fluids IJNFDW 0271-2091, in press.
22. MacDonald, I., Baines, M. J., Nichols, N. K., and, Samuels, P. G. (1997). " Analytical benchmark solutions for open-channel flows." J. Hydraul. Eng. JHEND8 0733-9429, 123 (11), 1041-1045. 10.1061/(ASCE)0733-9429(1997)123:11(1041)
23. Osher, S. (1984). " Convergence of generalized MUSCL schemes." SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. SJNAAM 0036-1429, 22 (5), 947-961. 10.1137/0722057
24. Osher, S., and, Solomon, F. (1982). " Upwind difference schemes for hyperbolic conservation laws." Math. Comput. MCMPAF 0025-5718, 38 (158), 339-374. 10.2307/2007275
25. Reed, W. H., and, Hill, T. R. (1973). " Triangular mesh methods for neutron transport equation." Technical Rep. No. LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, N.M.
26. Roe, P. L. (1981). " Approximate Riemann solvers, parameter vectors, and difference schemes." J. Comput. Phys. JCTPAH 0021-9991, 43 (2), 357-372. 10.1016/0021-9991(81)90128-5
27. Roe, P. L., and, Pike, J. (1984). " Efficient construction and utilization of approximate Riemann solutions." Proc., Computing Methods in Applied Science and Engineering, R. Glowinski and J. L. Lions., eds., North-Holland, Amsterdam, The Netherlands.
28. Shu, C.-W., and, Osher, S. (1988). " Efficient implementation of essentially non-oscillatory shock-capturing schemes." J. Comput. Phys. JCTPAH 0021-9991, 77 (2), 439-471. 10.1016/0021-9991(89)90222-2
29. Shu, C.-W., and, Osher, S. (1989). " Efficient implementation of essentially non-oscillatory shock-capturing schemes II." J. Comput. Phys. JCTPAH 0021-9991, 83 (1), 32-78. 10.1016/0021-9991(89)90222-2
30. Toro, E. F. (1992). " Riemann problems and the WAF method for solving the two dimensional shallow water equations." Philos. Trans. R. Soc. London PTRMAD 0962-8428, 338 (1649), 43-68. 10.1098/rsta.1992.0002
31. Toro, E. F. (1997). Riemann solvers and numerical methods for fluid dynamics, a practical introduction, Springer, Berlin.
32. Toro, E. F., Spruce, M., and, Speares, W. (1994). " Restoration of the contact surface in the HLL-Riemann solver." Shock Waves SHWAEN 0938-1287, 4 (1), 25-34. 10.1007/BF01414629
33. van Leer, B. (1979). " Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method." J. Comput. Phys. JCTPAH 0021-9991, 32 (1), 101-136. 10.1016/0021-9991(79)90145-1
34. Xing, Y., and, Shu, C.-W. (2006). " 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." Comm. Comp. Phys. ZZZZZZ 1815-2406, 1 (1), 101-135.
35. Zhou, T., Li, Y., and, Shu, C.-W. (2001). " Numerical comparison of WENO finite volume and Runge-Kutta discontinuous Galerkin methods." J. Sci. Comput. JSCOEB 0885-7474, 16 (2), 145-171. 10.1023/A:1012282706985