Central-upwind schemes for the Saint-Venant system
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 3, p. 397-425
We present one- and two-dimensional central-upwind schemes for approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutions in which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preserve this delicate balance with numerical schemes. Small perturbations of these states are also very difficult to compute. Our approach is based on extending semi-discrete central schemes for systems of hyperbolic conservation laws to balance laws. Special attention is paid to the discretization of the source term such as to preserve stationary steady-state solutions. We also prove that the second-order version of our schemes preserves the nonnegativity of the height of the water. This important feature allows one to compute solutions for problems that include dry areas.
DOI : https://doi.org/10.1051/m2an:2002019
Classification:  65M06,  35L65
Keywords: Saint-Venant system, shallow water equations, high-order central-upwind schemes, balance laws, conservation laws, source terms
@article{M2AN_2002__36_3_397_0,
     author = {Kurganov, Alexander and Levy, Doron},
     title = {Central-upwind schemes for the Saint-Venant system},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {3},
     year = {2002},
     pages = {397-425},
     doi = {10.1051/m2an:2002019},
     zbl = {1137.65398},
     mrnumber = {1918938},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2002__36_3_397_0}
}
Kurganov, Alexander; Levy, Doron. Central-upwind schemes for the Saint-Venant system. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 3, pp. 397-425. doi : 10.1051/m2an:2002019. http://www.numdam.org/item/M2AN_2002__36_3_397_0/

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