On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) no. 5, pp. 821-852.

This paper is concerned with the numerical approximations of Cauchy problems for one-dimensional nonconservative hyperbolic systems. The first goal is to introduce a general concept of well-balancing for numerical schemes solving this kind of systems. Once this concept stated, we investigate the well-balance properties of numerical schemes based on the generalized Roe linearizations introduced by [Toumi, J. Comp. Phys. 102 (1992) 360-373]. Next, this general theory is applied to obtain well-balanced schemes for solving coupled systems of conservation laws with source terms. Finally, we focus on applications to shallow water systems: the numerical schemes obtained and their properties are compared, in the case of one layer flows, with those introduced by [Bermúdez and Vázquez-Cendón, Comput. Fluids 23 (1994) 1049-1071]; in the case of two layer flows, they are compared with the numerical scheme presented by [Castro, Macías and Parés, ESAIM: M2AN 35 (2001) 107-127].

DOI : https://doi.org/10.1051/m2an:2004041
Classification : 65M99,  76B55,  76B70
Mots clés : nonconservative hyperbolic systems, well-balanced schemes, Roe method, source terms, shallow-water systems
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author = {Par\'es, Carlos and Castro, Manuel},
title = {On the well-balance property of {Roe's} method for nonconservative hyperbolic systems. {Applications} to shallow-water systems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {821--852},
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Parés, Carlos; Castro, Manuel. On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) no. 5, pp. 821-852. doi : 10.1051/m2an:2004041. http://www.numdam.org/articles/10.1051/m2an:2004041/

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