Influence of Total-Variation-Diminishing Slope Limiting on Local Discontinuous Galerkin Solutions of the Shallow Water Equations
Publication: Journal of Hydraulic Engineering
Volume 138, Issue 2
Abstract
Finite volume (FV) slope limiting is essential to stabilize discontinuous Galerkin (DG) solutions despite a number of side effects such as local loss of accuracy and increased run-time cost. These side effects have been experienced with DG solutions to the homogeneous system of conservation laws and are usually accepted as long as they do not affect the reliability of the numerical predictions and provide a better stability property. They have also led to the concept of localizing the slope-limiting process. When a model is applied to simulate the flow problems that necessitate the solution of the inhomogeneous shallow water equations (SWEs) with/without flooding and drying, the slope-limiting routine can have extra adverse effects on the local DG framework. These effects, causes, and implications on DG solutions to the SWEs have not yet received adequate attention, and a full investigation is therefore needed. With the aim of improving a second-order Runge-Kutta discontinuous Galerkin (RKDG2) SWE solver, the effects of FV slope limiting on the RKDG2 flow solutions were analyzed. Three versions of the RKDG2 scheme were configured: the first and second versions were respectively associated with the local limiting (LL) and global limiting (GL) processes, which both use limiter-free linear front tracking (LFT) to trace a wet/dry front, and the third was implemented with the local limiting process but with constant front-tracking (CFT) of the wet/dry front. Selected hypothetical and practical hydraulic tests were investigated to demonstrate the effects of different limiting processes and front-tracking techniques.
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Acknowledgments
This work was funded by the UK Engineering and Physical Sciences Research Council (EPSRC) through Grant EPSRC-GBEP/F030177/1.
References
Aureli, F., Maranzoni, A., Mignosa, P., and Ziveri, C. (2008). “A weighted surface-depth gradient method for the numerical integration of the 2D shallow water equations with topography.” Adv. Water Resour., 31(7), 962–974.
Begnudelli, L., and Sanders, B. F. (2007). “Simulation of the St. Francis dam-break flood.” J. Eng. Mech., 133(11), 1200–1212.
Bokhove, O. (2005). “Flooding and drying in discontinuous Galerkin finite element method for shallow water flows.” J. Sci. Comput., 22–23(1), 47–82.
Bradford, S. F., and Sanders, B. F. (2002). “Finite volume model for shallow water flooding of arbitrary topography.” J. Hydraul. Eng., 128(3), 289–298.
Brufau, P., García-Navarro, P., and Vázquez-Cendón, M. E. (2004). “Zero mass error using unsteady wetting-drying conditions in shallow flows over dry irregular topography.” Int. J. Numer. Method Fluids, 45(10), 1047–1082.
Bunya, S., Kubatko, E. J., Westerink, J. J., and Dawson, C. (2009). “A wetting and drying treatment for the Runge-Kutta discontinuous Galerkin solution to the shallow water equations.” Comput. Meth. Appl. Mech. Eng., 198(17–20), 1548–1562.
Cockburn, B. (2003). “Discontinuous Galerkin methods.” Z. Angew. Math. Mech., 83(11), 731–754.
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. Method Fluids, 58(1), 1–25.
Kesserwani, G., and Liang, Q. (2010). “Well-balanced RKDG2 solutions to the shallow water equations over irregular domains with wetting and drying.” Comput. Fluids, 39(10), 2040–2050.
Kesserwani, G., Liang, Q., Mosé, R., and Vazquez, J. (2010). “Well-balancing issues related to the RKDG2 scheme for the shallow water equations.” Int. J. Numer. Method Fluids, 62(4), 428–448.
Krivodonova, L., Xin, J., Remacle, J. F., Chevaugeon, N., and Flaherty, J. E. (2004). “Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws.” Appl. Numer. Math., 48(3–4), 323–338.
Liang, Q. (2010). “Flood simulation using a well-balanced shallow flow model.” J. Hydraul. Eng., 136(9), 669–675.
Liang, Q., and Marche, F. (2009). “Numerical resolution of well-balanced shallow water equations with complex source terms.” Adv. Water Resour., 32(6), 873–884.
Meselhe, E. A., Sotiropoulos, F., and Holly, F. M. Jr. (1997). “Numerical simulation of transcritical flow in open Channels.” J. Hydraul. Eng., 123(9), 774–783.
Qiu, J., and Shu, C.-W. (2005). “A comparison of troubled-cell indicators for Runge-Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters.” SIAM J. Sci. Comput., 27(3), 995–1013.
Sanders, B. F., and Bradford, S. F. (2006). “Impact of limiters on accuracy of high-resolution flow and transport models.” J. Eng. Mech., 132(1), 87–98.
Toro, E. F. (2001). Shock-capturing methods for free-surface shallow flows, Wiley, Chichester, UK.
Toro, E. F., and García-Navarro, P. (2007). “Godunov-type methods for free-surface shallow flows: A review.” J. Hydraul. Res., 45(6), 736–751.
Xing, Y., Zhang, X., and Shu, C.-W. (2010). “Positivity-preserving high-order well-balanced discontinuous Galerkin methods for the shallow water equations.” Adv. Water Resour., 33(12), 1476–1493.
Yang, M., and Wang, Z. J. (2009). “A parameter-free generalized moment limiter for high-order methods on unstructured grids.” Adv. Appl. Math. Mech., 1(4), 451–480.
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© 2012 American Society of Civil Engineers.
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Received: Feb 12, 2010
Accepted: Jul 22, 2011
Published online: Jul 25, 2011
Published in print: Feb 1, 2012
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