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Journal of Scientific Computing

Erschienen in:

01.08.2013

A Well-Balanced Reconstruction of Wet/Dry Fronts for the Shallow Water Equations

verfasst von: Andreas Bollermann, Guoxian Chen, Alexander Kurganov, Sebastian Noelle

Erschienen in: Journal of Scientific Computing | Ausgabe 2/2013

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Abstract

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.

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Nächster Artikel The Compact Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity

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Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R., Perthame, B.: A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25, 2050–2065 (2004)MathSciNetMATHCrossRef

Bollermann, A., Noelle, S., Lukáčová-Medvid’ová, M.: Finite volume evolution Galerkin methods for the shallow water equations with dry beds. Commun. Comput. Phys. 10, 371–404 (2011)MathSciNet

de Saint-Venant, A.J.C.: Théorie du mouvement non-permanent des eaux, avec application aux crues des rivière at à l’introduction des warées dans leur lit. C. R. Acad. Sci. Paris 73, 147–154 (1871)MATH

Gallardo, J.M., Parés, C., Castro, M.: On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas. J. Comput. Phys. 2227, 574–601 (2007)CrossRef

Gallouët, T., Hérard, J.-M., Seguin, N.: Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32, 479–513 (2003)MathSciNetMATHCrossRef

Godlewski, E., Raviart, P.-A.: Numerical approximation of hyperbolic systems of conservation laws. Springer, New York (1996)MATH

Gottlieb, S., Shu, C.-W., Tadmor, E.: High order time discretization methods with the strong stability property. SIAM Rev. 43, 89–112 (2001)MathSciNetMATHCrossRef

Jin, S.: A steady-state capturing method for hyperbolic system with geometrical source terms. M2AN Math. Model. Numer. Anal. 35, 631–645 (2001)MathSciNetMATHCrossRef

Jin, S., Wen, X.: Two interface-type numerical methods for computing hyperbolic systems with geometrical source terms having concentrations. SIAM J. Sci. Comput. 26, 2079–2101 (2005)MathSciNetMATHCrossRef

10.

Kröner, D.: Numerical schemes for conservation laws. Wiley, Chichester (1997)MATH

11.

Kurganov, A., Levy, D.: Central-upwind schemes for the Saint-Venant system. M2AN Math. Model. Numer. Anal. 36, 397–425 (2002)MathSciNetMATHCrossRef

12.

Kurganov, A., Petrova, G.: A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun. Math. Sci. 5(1), 133–160 (2007)MathSciNetMATH

13.

LeVeque, R.: Finite volume methods for hyperbolic problems. Cambridge texts in applied mathematics. Cambridge University Press, Cambridge (2002)CrossRef

14.

LeVeque, R.J.: Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146, 346–365 (1998)MathSciNetMATHCrossRef

15.

Liang, Q., Marche, F.: Numerical resolution of well-balanced shallow water equations with complex source terms. Adv. Water Resour. 32(6), 873–884 (2009)CrossRef

16.

Lie, K.-A., Noelle, S.: On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws. SIAM J. Sci. Comput. 24, 1157–1174 (2003)MathSciNetMATHCrossRef

17.

Morris, M.: CADAM: concerted action on dam break modelling—final report. HR Wallingford, Wallingford (2000)

18.

Nessyahu, H., Tadmor, E.: Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463 (1990)MathSciNetMATHCrossRef

19.

Noelle, S., Pankratz, N., Puppo, G., Natvig, J.: Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213, 474–499 (2006)MathSciNetMATHCrossRef

20.

Perthame, B., Simeoni, C.: A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38, 201–231 (2001)MathSciNetMATHCrossRef

21.

Ricchiuto, M., Bollermann, A.: Stabilized residual distribution for shallow water simulations. J. Comput. Phys. 228, 1071–1115 (2009)MathSciNetMATHCrossRef

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)

23.

Russo, G.: Central schemes for conservation laws with application to shallow water equations. In: Rionero, S., Romano, G. (eds.) Trends and applications of mathematics to mechanics: STAMM 2002, pp. 225–246. Springer Italia SRL, Berlin (2005)CrossRef

24.

Singh, J., Altinakar, M.S., Ding, Y.: Two-dimensional numerical modeling of dam-break flows over natural terrain using a central explicit scheme. Adv. Water Resour. 34, 1366–1375 (2011)CrossRef

25.

Sweby, P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21, 995–1011 (1984)MathSciNetMATHCrossRef

26.

Synolakis, C.E.: The runup of long waves. Ph.D. thesis, California Institute of Technology (1986)

27.

Synolakis, C.E.: The runup of solitary waves. J. Fluid Mech. 185, 523–545 (1987)MATHCrossRef

28.

Tai, Y.C., Noelle, S., Gray, J.M.N.T., Hutter, K.: Shock-capturing and front-tracking methods for granular avalanches. J. Comput. Phys. 175, 269–301 (2002)MATHCrossRef

29.

van Leer, B.: Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136 (1979)CrossRef

30.

Xing, Y., Shu, C.-W.: High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208, 206–227 (2005)MathSciNetMATHCrossRef

31.

Xing, Y., Shu, C.-W.: 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. Commun. Comput. Phys. 1, 100–134 (2006)MATH

32.

Xing, Y., Zhang, X., Shu, C.W.: Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv. Water Resour. 33(12), 1476–1493 (2010)CrossRef

Titel: A Well-Balanced Reconstruction of Wet/Dry Fronts for the Shallow Water Equations
verfasst von: Andreas Bollermann
Guoxian Chen
Alexander Kurganov
Sebastian Noelle
Publikationsdatum: 01.08.2013
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 2/2013
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-012-9677-5

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