A Well-Balanced Reconstruction of Wet/Dry Fronts for the Shallow Water Equations
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In this paper, we construct a well-balanced, positivity preserving finite volume scheme for the shallow water equations based on a continuous, piecewise linear discretization of the bottom topography. The main new technique is a special reconstruction of the flow variables in wet–dry cells, which is presented in this paper for the one dimensional case. We realize the new reconstruction in the framework of the second-order semi-discrete central-upwind scheme from (Kurganov and Petrova, Commun. Math. Sci., 5(1):133–160, 2007). The positivity of the computed water height is ensured following (Bollermann et al., Commun. Comput. Phys., 10:371–404, 2011): The outgoing fluxes are limited in case of draining cells.
KeywordsHyperbolic systems of conservation and balance laws Saint-Venant system of shallow water equations Finite volume methods Well-balanced schemes Positivity preserving schemes Wet/dry fronts
The first ideas for this work were discussed by the authors at a meeting at the “Mathematisches Forschungsinstitut Oberwolfach”. The authors are grateful for the support and inspiring atmosphere there. The research of A. Kurganov was supported in part by the NSF Grant DMS-1115718 and the ONR Grant N000141210833. The research of A. Bollermann, G. Chen and S. Noelle was supported by DFG Grant NO361/3-1 and No361/3-2. G. Chen is partially supported by the National Natural Science Foundation of China (No. 11001211, 51178359).
- 17.Morris, M.: CADAM: concerted action on dam break modelling—final report. HR Wallingford, Wallingford (2000)Google Scholar
- 22.Russo, G.: Central schemes for balance laws. In: Hyperbolic problems: theory, numerics, applications, vols. I, II (Magdeburg, 2000), pp. 821–829. International Series of Numerical Mathematics vols. 140, 141, Birkhäuser, Basel (2001)Google Scholar
- 26.Synolakis, C.E.: The runup of long waves. Ph.D. thesis, California Institute of Technology (1986)Google Scholar