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Godunov method for nonconservative hyperbolic systems. (English) Zbl 1124.65077
The paper is concerned with the numerical approximation of Cauchy problems for one-dimensional nonconservative hyperbolic systems. For the definition of the weak solutions of the system, the theory: an interpretation of the nonconservative products as Borel measures is given, based on the choice of a family of paths drawn in the phase space, is used. Even if the family of paths can be chosen arbitrarily, it is natural to require this family to satisfy some hypotheses concerning the relation of the paths with the integral curves of the characteristic fields.
The investigation of the implication of three basic hypotheses of this nature is the main purpose of this work. It is also shown that when the family of paths satisfies these hypotheses, Godunov methods can be written in a natural form that generalizes their classical expression for systems of conservation laws. The well-balance properties of these methods are also studied. The consistency of the numerical scheme with the definition of weak solutions is finally proved.

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35L60 First-order nonlinear hyperbolic equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
HLLE
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