Abstract
We consider the numerical approximation of the shallow–water equations with non–flat topography. We introduce a new topography discretization that makes all schemes to be well–balanced and robust. At the discrepancy with the well–known hydrostatic reconstruction, the proposed numerical procedure does not involve any cut–off. Moreover, the obtained scheme is able to deal with dry areas. Several numerical benchmarks are performed to assert the interest of the method.
Keywords
MSC2010: 65M12, 76M12, 35L65
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
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.Comp., 25, 2050–2065 (2004).
Bermudez A., Vazquez–Cendon M.E., Upwind Methods for Hyperbolic Conservation Laws with Source Terms, Computers and Fluids. 23 1049–1071 (1994).
Berthon C., Coquel F.: Nonlinear projection methods for multi–entropies Navier–Stokes systems, Math. Comput., 76, 1163–1194 (2007).
Berthon C., Dubois J., Dubroca B., Nguyen–Bui T.H., Turpault R.: A Free Streaming Contact Preserving Scheme for the M1 Model, Adv. Appl. Math. Mech., 3, 259–285 (2010).
Berthon C., Marche F.: A positive well–balanced VFRoe–ncv scheme for non–homogeneous shallow–water equations, ISCM–EPMESC proceedings, AIP Conference Proceedings, 1495–1500 (2010).
Bouchut F.: Non–linear stability of finite volume methods for hyperbolic conservation laws and well–balanced schemes for sources, Frontiers in Mathematics, Birkhauser, 2004.
Castro M., LeFloch P., Munoz–Ruiz M.L., Parés C.: Why many theories of shock waves are necessary: convergence error in formally path–consistent schemes, J. Comput. Phys. 227, 8107âĂŞ-8129 (2008).
Dal Maso G., LeFloch P., Murat F., Definition and weak stability of a non conservative product, J. Math. Pures Appl., 74, 483–548 (1995).
Delestre O., Simulation du ruissellement d’eau de pluie sur des surfaces agricoles, PhD Thesis, University of Orléans, http://tel.archives-ouvertes.fr/tel-00531377/en (2010).
Gallouët T., Hérard J.M., Seguin N.: Some recent Finite Volume schemes to compute Euler equations using real gas EOS, Int J. Num. Meth. Fluids, 39, 1073–1138 (2002).
Gallouet T., Hérard J.M., Seguin N.: Some approximate Godunov schemes to compute shallow–water equations with topography, Computers and Fluids, 32, 479–513 (2003).
Gallouet T., Hérard J.M., Seguin N.: On the use of some symetrizing variables to deal with vacuum, Calcolo, 40, 163–194 (2003).
Greenberg J.M., Leroux A.Y.: A well–balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33, 1–16 (1996).
Hou T.Y., LeFloch P.: Why non–conservative schemes converge to wrong solutions: error analysis, Math. Comput., 206, pp. 497–530 (1994).
Jin S.: A steady–state capturing method for hyperbolic systems with geometrical source terms, M2AN Math. Model. Numer. Anal., 35, 631–645 (2001).
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).
Jin S., Wen X.: An efficient method for computing hyperbolic systems with geometrical source terms having concentrations, Special issue dedicated to the 70th birthday of Professor Zhong–Ci Shi. J. Comput. Math., 22, 230–249 (2004).
Marche F., Bonneton P., Fabrie P., Seguin N.: Evaluation of well–balanced bore–capturing schemes for 2D wetting and drying processes, Int. J. Numer. Meth. Fluids, 53, 867–894 (2007).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Berthon, C., Foucher, F. (2011). Hydrostatic Upwind Schemes for Shallow–Water Equations. In: Fořt, J., Fürst, J., Halama, J., Herbin, R., Hubert, F. (eds) Finite Volumes for Complex Applications VI Problems & Perspectives. Springer Proceedings in Mathematics, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20671-9_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-20671-9_11
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-20670-2
Online ISBN: 978-3-642-20671-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)