Discontinuous Galerkin Method for 1D Shallow Water Flow in Nonrectangular and Nonprismatic Channels
Publication: Journal of Hydraulic Engineering
Volume 138, Issue 3
Abstract
A total variation diminishing Runge-Kutta discontinuous Galerkin finite element method for the solution of one-dimensional (1D) shallow water flow equations for natural channels is presented. The hydrostatic pressure force and the wall pressure force terms are combined to simplify the calculations and prevent unphysical flow attributable to improper treatment of the bottom slope term. The treatment of the combined term that appropriately accounts for the momentum flux is given. HLL and Roe Riemann solvers are assessed for the mass and momentum flux terms. Numerical tests are conducted using prismatic rectangular and nonrectangular channels as well as non prismatic channels and natural channel for dam break, supercritical flow, transcritical flow, and dry-bed problems. Slope limiters based on flow cross section area, water surface, and water depth are evaluated. The tests show that HLL and Roe solvers provide similar accuracy. However, the slope limiter based on flow area provides more accurate solutions for tests in nonrectangular and natural channels.
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References
Bellos, C., Soulis, V., and Sakkas, J. (1992). “Experimental investigation of two-dimensional dam-break induced flows.” J. Hydraul. Res., 30(1), 47–63.
Bokhove, O. (2005). “Flooding and drying in discontinuous Galerkin finite-element discretizations of shallow-water equations. Part 1: one dimension.” J. Sci. Comput., 22-23(1–3), 47–82.
Catella, M., Paris, E., and Solari, L. (2008). “Conservative scheme for numerical modeling of flow in natural geometry.” J. Hydraul. Eng., 134(6), 736–748.
Cockburn, B. (2001). “Discontinuous Galerkin methods for convection dominated problems.” Lecture Notes in Computational Science and Engineering, Vol. 9, Springer, Berlin, 69–224.
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., 54(190), 545–581.
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., 84(1), 90–113.
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., 52(186), 411–435.
Cockburn, B., and Shu, C. W. (1998). “The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems.” J. Comput. Phys., 141(2), 199–224.
Cunge, J. A., Holly, F. M., and Verwey, A. (1980). Practical Aspects of Computational River Hydraulics, Pitman, London.
Dolejši, V. (2010). “On the solution of linear algebraic systems arising from the semi-implicit DGFE discretization of the compressible Navier-Stokes equations.” Kybernetika, 46(2), 260–280.
Einfeldt, B. (1988). “On Godunov-type methods for gas dynamics.” SIAM J. Numer. Anal., 25(2), 294–318.
Ern, A., Piperno, S., and Djadel, K. (2008). “A well-balanced Runge-Kutta discontinuous Galerkin method for the shallow-water equations with flooding and drying.” Int. J. Numer. Methods Fluids, 58(1), 1–25.
Fagherazzi, S., Rasetarinera, P., Hussaini, M. Y., and Furbish, D. (2004). “Numerical solution of the dam-break problem with a discontinuous Galerkin method.” J. Hydraul. Eng., 130(6), 532–539.
Fraccarollo, L., and Toro, E. F. (1995). “Experimental and numerical assessment of the shallow water model for two-dimensional dam-break type problems.” J. Hydraul. Res., 33(6), 843–846.
Garcia-Navarro, P., and Vázquez-Cendón, M. (2000). “On numerical treatment of the source terms in the shallow water equations.” Computers & Fluids, 29(8), 951–979.
Godunov, S. K. (1959). “A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics.” Mat. Sb., 47(3), 271–306.
Gottlieb, S., and Shu, C. W. (1998). “Total variation diminishing Runge-Kutta schemes.” Math. Comput., 67(221), 73–85.
Harten, A., Lax, P. D., and van Leer, B. (1983). “On upstream differencing and Godunov-type schemes for hyperbolic conservation laws.” SIAM Rev., 25(1), 35–61.
Henderson, F. M. (1966). Open Channel Flow, McGraw-Hill, New York.
Kesserwani, G., Ghostine, R., Vazquez, J., Ghenaim, A., and Mosé, R. (2008a). “Application of a second-order Runge-Kutta discontinuous Galerkin scheme for the shallow water equations with source terms.” Int. J. Numer. Methods Fluids, 56(7), 805–821.
Kesserwani, G., Ghostine, R., Vazquez, J., Ghenaim, A., and Mosé, R. (2008b). “One-dimensional simulation of supercritical flow at a confluence by means of a nonlinear junction model applied with the RKDG2 method.” Int. J. Numer. Methods Fluids, 57(12), 1695–1708.
Kesserwani, G., Ghostine, R., Vazquez, J., Ghenaim, A., and Mosé, R. R. (2008c). “Riemann solvers with Runge-Kutta discontinuous Galerkin schemes for the 1D shallow water equations.” J. Hydraul. Eng., 134(2), 243–355.
Kesserwani, G., and Liang, Q. (2011). “A conservative high-order discontinuous Galerkin method for the shallow water equations with arbitrary topography.” Int. J. Numer. Methods Eng., 86(1), 47–69.
Kesserwani, G., Liang, Q., Vazquez, J., and Mosé, R. (2010). “Well-balancing issues related to the RKDG2 scheme for the shallow water equations.” Int. J. Numer. Methods Fluids, 62(4), 428–448.
Kesserwani, G., Mosé, R., Vazquez, J., and Ghenaim, A. (2009). “A practical implementation of high-order RKDG models for the 1D open-channel flow equations.” Int. J. Numer. Methods Fluids, 59(12), 1389–1409.
Khalifa, A. M. (1980). “Theoretical and experimental study of the radial hydraulic jump.” Ph.D. thesis, Univ. of Windsor, Windsor, ON, Canada.
Khan, A. A. (2000). “Modeling flow over an initially dry bed.” J. Hydraul. Res., 38(5), 383–388.
Li, B. Q. (2006). Discontinuous Finite Element in Fluid Dynamics and Heat Transfer, Springer, London.
Nujić, M. (1995). “Efficient implementation of non-oscillatory schemes for the computation of free-surface flows.” J. Hydraul. Res., 33(1), 101–111.
Perthame, B., and Simeoni, C. (2001). “A kinetic scheme for the Saint-Venant system with a source term.” Calcolo, 38(4), 201–231.
Reed, W. H., and Hill, T. (1973). “Triangular mesh method for the neutron transport equation.” Los Alamos Report LA-UR-73-479.
Roe, P. (1981). “Approximate Riemann solvers, parameter vectors, and difference schemes.” J. Comput. Phys., 43(2), 357–372.
Sanders, B. F. (2001). “High-resolution and non-oscillatory solution of the St. Venant equations in non-rectangular and non-prismatic channels.” J. Hydraul. Res., 39(3), 321–330.
Sanders, B. F., Jaffe, D. A., and Chu, A. K. (2003). “Discretization of integral equations describing flow in nonprismatic channels with uneven beds.” J. Hydraul. Eng., 129(3), 235–244.
Schwanenberg, D., Kiem, R., and Köngeter, J. (2000). “Discontinuous Galerkin Method for the shallow water equations.” Discontinuous Galerkin Methods, B, G. E. Krniadakis and C. W. Shu, eds., Springer, Heidelberg, 289–309.
Thacker, W. C. (1981). “Some exact solutions to the nonlinear shallow-water wave equations.” J. Fluid Mech., 107, 499–508.
Toro, E. (1989). “A weighted average flux method for hyperbolic conservation laws.” Proc. R. Soc. London, 423(1865), 401–418.
Toro, E., Spruce, M., and Speares, W. (1994). “Restoration of the contact surface in the HLL-Riemann solver.” Shock Waves, 4(1), 25–34.
United States Army Corps of Engineers (USACE). (1960). “Floods resulting from suddenly breached dams.” Misc. Paper No. 2-374, Report 1: Conditions of Minimum Resistance, Waterways Experiment Station, Vicksburg, MS.
Xing, Y., Zhang, X., and Shu, C. (2010). “Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations.” Adv. Water Resour., 33(12), 1476–1493.
Ying, X., Khan, A. A., and Wang, S. S. Y. (2004). “Upwind conservative scheme for the Saint Venant equations.” J. Hydraul. Eng., 130(10), 977–987.
Zhou, J. G., Causon, D. M., Mingham, C. G., and Ingram, D. M. (2001). “The surface gradient method for the treatment of source terms in the shallow-water equations.” J. Comput. Phys., 168(1), 1–25.
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Information
Published In
Journal of Hydraulic Engineering
Volume 138 • Issue 3 • March 2012
Pages: 285 - 296
Copyright
© 2012 American Society of Civil Engineers.
History
Received: Feb 7, 2011
Accepted: Aug 24, 2011
Published online: Aug 26, 2011
Published in print: Mar 1, 2012
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