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Discontinuous Galerkin Method for 1D Shallow Water Flow with Water Surface Slope Limiter
Abstract:A water surface slope limiting scheme is tested and compared with the water depth slope limiter for the solution of one dimensional shallow water equations with bottom slope source term. Numerical schemes based on the total variation diminishing Runge- Kutta discontinuous Galerkin finite element method with slope limiter schemes based on water surface slope and water depth are used to solve one-dimensional shallow water equations. For each slope limiter, three different Riemann solvers based on HLL, LF, and Roe flux functions are used. The proposed water surface based slope limiter scheme is easy to implement and shows better conservation property compared to the slope limiter based on water depth. Of the three flux functions, the Roe approximation provides the best results while the LF function proves to be least suitable when used with either slope limiter scheme.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1331823Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 2284
 J. S. Wang, H. G. Ni and Y. S. He, "Finite-difference TVD scheme for computation of dam-break problems," J. Hydr. Engng., vol. 126, no. 4, pp. 253-262, 2000.
 G. F. Lin, J. S. Lai and W. D. Guo, "Finite-volume component-wise TVD schemes for 2D shallow water equations," Adv. Water Resour., vol. 26, no. 8, pp. 861-873, 2003.
 T. J. R. Hughes, W. K. Liu and A. Brooks, "Finite element analysis of incompressible viscous flows by the penalty function formulation," J. Comput. Phys., vol. 30, no. 1, pp. 1-60, 1979.
 O. C. Zienkiewicz and P. Ortiz, "A split-characteristic based finite element model for the shallow equations," Int. J. Numer. Methods Fluids, vol. 20, no. 8-9, pp. 1061-1080, 1995.
 T. Arbogast and M. F. Wheeler, "A characteristics-mixed finite element method for advection-dominated transport problems," SIAM J. Numer. Anal., vol. 32, no. 2, pp. 404-424, 1995.
 W. H. Reed and T. Hill, "Triangular Mesh Method for the Neutron Transport Equation," Los Alamos Report, LA-UR-73-479, 1973.
 B. Cockburn and C. W. Shu, "TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework," Math. Comp., vol. 52, pp. 411-435, 1989.
 B. Cockburn, S. Y. Lin and C. W. Shu, "TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems," J. Comput. Phys, vol. 84, no. 1, pp. 90-113, 1989.
 B. Cockburn, S. Hou and C. W. Shu, "The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV: The multidimensional case," Math. Comp., vol. 54, pp. 545-581, 1990.
 B. Cockburn and C. W. Shu, "The Runge-Kutta discontinuous Galerkin finite element method for conservation laws V: Multidimensional systems," J. Comput. Phys., vol. 141, no. 2, pp. 199-224, 1998.
 B. Q. Li, "Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer," Springer Verlag, 2006.
 D. Schwanenberg and J. Köngeter, "A discontinuous Galerkin method for the shallow water equations with source terms," Lecture Notes in Computational Science and Engineering, Springer, Berlin, vol. 11, pp. 419-424, 2000.
 D. Schwanenberg and M. Harms, "Discontinuous Galerkin finiteelement method for transcritical two-dimensional shallow water flows," J. Hydr. Engng., vol. 130, no. 5, pp. 412-421, 2004.
 V. Aizinger and C. Dawson, "A discontinuous Galerkin method for twodimensional flow and transport in shallow water," Adv. Water Resour., vol. 25, no. 1, pp. 67-84, 2002.
 C. Dawson and V. Aizinger, "A discontinuous Galerkin method for three-dimensional shallow water equations," J. Sci. Comput., vol. 22, no. 1, pp. 245-267, 2005.
 E. J. Kubatko, J. J. Westerink and C. Dawson, "hp discontinuous Galerkin methods for advection dominated problems in shallow water flow," Comput. Methods Appl. Mech. Eng., vol. 196, no, 1-3, pp. 437- 451, 2006.
 A. Harten, P. D. Lax and B. van Leer, "On upstream differencing and Godunov-type schemes for hyperbolic conservation laws," SIAM Rev., vol. 25, no. 1, pp. 35-61, 1983.
 P. Roe, "Approximate Riemann solvers, parameter vectors, and difference schemes," J. Comput. Phys., vol. 43, no. 2, pp. 357-372, 1981.
 P. Tassi, O. Bokhove and C. Vionnet, "Space discontinuous Galerkin method for shallow water flowsÔÇökinetic and HLLC flux, and potential vorticity generation," Adv. Water Resour., vol. 30, no. 4, pp. 998-1015, 2007.
 S. Gottlieb and C. W. Shu, "Total variation diminishing Runge-Kutta schemes," Math. Comp., vol. 67, 73-85, 1998.
 J. G. Zhou, D. M. Causon, C. G. Mingham and D. M. Ingram, "The surface gradient method for the treatment of source terms in the shallowwater equations," J. Comput. Phys., vol. 168, no. 1, pp. 1-25, 2001.
 X. Ying, A. A. Khan and S. S. Y. Wang, "Upwind conservative scheme for the Saint Venant equations," J. Hydr. Engng., vol. 130, no. 4, pp. 977-987, 2004.
 M. Catella, E. Paris and L. Solari, "Conservative scheme for numerical modeling of flow in natural geometry," J. Hydr. Engng., vol. 134, no. 6, pp. 736-748, 2008.
 B. Cockburn, "Discontinuous Galerkin Methods for Convection Dominated Problems," Lecture Notes in Computational Science and Engineering, Springer, Berlin, vol. 9, pp. 69-224, 2001.