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High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. (English) Zbl 1096.65082
This paper is concerned with the construction of high order schemes for the numerical solution of one-dimensional nonconservative hyperbolic systems. Based on the use of a first order Roe scheme and weighted essentially nonoscillatory reconstruction of states, the authors give the general expression of a well-balanced high order scheme. For such a general expression, once obtained, particular schemes can be deduced for any system of the form discussed in the paper, where the numerical treatment of source and coupling terms is automatically derived. The authors also study the well-balanced properties of the resulting schemes and apply the schemes to shallow-water systems to verify the well-balanced property numerically.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76M12 Finite volume methods applied to problems in fluid mechanics
35L65 Hyperbolic conservation laws
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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