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On the convergence and well-balanced property of path-conservative numerical schemes for systems of balance laws. (English) Zbl 1230.65102
This article is concerned with the numerical approximation of Cauchy problems for one-dimensional systems of conservation laws with source terms or balance laws. The authors prove that, for systems of balance laws with sufficiently smooth right hand side, a Lax-Wendroff type result can be obtained for path-conservative numerical schemes assuming the same hypothesis of convergence used in the classical Lax-Wendroff theorem for conservative problems. Together with the hypothesis concerning the convergence of the approximations, some assumptions both on the chosen family of paths and on the numerical scheme are also required. Moreover, this paper presents some general results showing that some further assumptions have to be imposed to the family of paths, in order to obtain well-balanced schemes.

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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