ESAIM: Mathematical Modelling and Numerical Analysis

Table of Contents - September 2004 - Volume 38 - Issue05

Research Article

On the well-balance property of Roe's method for nonconservative hyperbolic systems. applications to shallow-water systems

Parés, Carlosa1 and Castro, Manuela1

a1 Dpto. Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos s/n, 29080-Málaga, Spain. grupo@anamat.cie.uma.es.

Abstract

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].

(Received November 3 2003)

(Revised June 9 2004)

(Online publication October 15 2004)

Key Words:

  • Nonconservative hyperbolic systems;
  • well-balanced schemes;
  • Roe method;
  • source terms;
  • shallow-water systems.

Mathematics Subject Classification:

  • 65M99;
  • 76B55;
  • 76B70
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