Boundary value problem for a strictly hyperbolic system related with the Godunov scheme. (Conditions aux limites pour un système strictement hyperbolique fournies, par le schéma de Godunov.) (French) Zbl 0873.65087

The paper deals with the numerical approximation of initial boundary value problems in the quarter plane, \(x>0\), \(t>0\), by means of the Godunov scheme. Under some classical assumptions - strict hyperbolicity, noncharacteristic boundary, existence of a strictly convex entropy - as well as a technical one - that is satisfied in most interesting cases - , the authors show that, for compatible initial and boundary data, the numerical solution converges, as the mesh size goes to zero, to the local-in-time smooth exact solution given by the theory of T.-T. Li and W.-C. Yu [Boundary value problems for quasilinear hyperbolic systems (1985; Zbl 0627.35001)]. The proof of the convergence, which holds - provided that the initial data are constant outside a bounded interval - in \(L^\infty(0,T;L^2(0,+\infty))\), \(T>0\) small enough, is based on entropy estimates. Some numerical experiments concerning the Lagrangian isentropic gas dynamics are provided. They show the existence of a boundary layer, as expected, in view of the viscous approximation analysis previously made by M. Gisclon [J. Math. Pures Appl., IX. Sér. 75, No. 5, 485-508 (1996; Zbl 0869.35061)] (see also M. Gisclon and D. Serre [C. R. Acad. Sci., Paris, Sér. I 319, No. 4, 377-382 (1994; Zbl 0808.35075)]).
Reviewer: S.Benzoni (Lyon)


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76M20 Finite difference methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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