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On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems. (English) Zbl 1130.76325
Summary: 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 I. Toumi [J. Comput. Phys. 102, No. 2, 360–373 (1992; Zbl 0783.65068)]. 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 [A. Bermúdez and M. E. Vázquez-Cendón, Comput. Fluids 23, No. 8, 1049–1071 (1994; Zbl 0816.76052)]; in the case of two layer flows, they are compared with the numerical scheme presented by M. Castro and C. Parés [ M2AN, Math. Model. Numer. Anal. 35, No. 1, 107–127 (2001; Zbl 1094.76046)].

MSC:
76B55 Internal waves for incompressible inviscid fluids
76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L45 Initial value problems for first-order hyperbolic systems
35L60 First-order nonlinear hyperbolic equations
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