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An asymptotic expansion for the solution of the generalized Riemann problem. I: General theory. (English) Zbl 0679.35064
The following generalized Riemann problem for nonlinear hyperbolic systems of conservation laws is considered: \[ u_ t+f(x,t,u)_ x=g(x,t,u),\quad x\in R,\quad t>0,\quad u\in R^ p, \] u(x,0) is smooth for \(x<0\) and \(x>0\). An entropy solution of this problem is found in the form of an asymptotic expansion in time of the type \[ v(z,t)=u(zt,t),\quad v(z,t)=v_ 0(z)+tv_ 1(z)+...+t^ kv_ k(z)+.... \] An explicit method of construction of this asymptotic expansion is given and an approximate solution is constructed in a suitable way from the truncated expansion. \(L^ 1\)-bound for the error of such an approximation is given.
Reviewer: A.Doktor

MSC:
35L65 Hyperbolic conservation laws
35A35 Theoretical approximation in context of PDEs
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References:
[1] Ben-Artzi, M.; Falcovitz, J., A second order Godunov-type scheme for compressible fluid dynamics, J. Comp. Phys., Vol. 55, 1-32, (1984) · Zbl 0535.76070
[2] Ben-Artzi, M.; Falcovitz, J., An upwind second-order scheme for compressible duct flows, S.I.A.M. J. Sci. Stat. Comp., Vol. 7, 744-768, (1986) · Zbl 0594.76057
[3] Bourgeade, A.; Le Floch, Ph.; Raviart, P. A., Approximate solution of the generalized Riemann problem and applications, Actes du Congrès Hyperbolique of Saint-Etienne (France), (1988), Springer-Verlag, (to appear) · Zbl 0703.35106
[4] A. Bourgeade, Ph. Le Floch and P. A. Raviart, An Asymptotic Expansion for the Solution of the Generalized Riemann Problem. Part. II: Application to the Gas Dynamics Equations (to appear). · Zbl 0703.35106
[5] Glimm, J.; Marshall, G.; Plohr, B., A generalized Riemann problem for quasi-one-dimensional gas flows, Adv. Appl. Maths., Vol. 5, 1-30, (1984) · Zbl 0566.76056
[6] Harabetian, E., A Cauchy-kowalevky theorem for strictly hyperbolic systems of conservation laws with piecewise analytic initial data, Ph. D. dissertation, (1984), University of California Los Angeles
[7] Le Floch, Ph.; Raviart, P. A., Un développement asymptotique pour la solution d’un problème de Riemann généralisé, C.R. Acad. Sci. Paris, T. 304, Séries I, n^{o} 4, (1987) · Zbl 0619.35074
[8] Li Tatsien and Yu Wenci, Boundary Value Problem for Quasilinear Hyperbolic Systems, Duke University Mathematics Series, 1985.
[9] Liu, T. P., Quasilinear hyperbolic systems, Comm. Math. Phys., Vol. 68, 141-142, (1979)
[10] Smoller, J., Shock waves and reaction-diffusion equations, (1983), Springer-Verlag New York · Zbl 0508.35002
[11] Van Leer, B., Towards the ultimate conservative difference scheme, V, J. Comp. Phys., Vol. 32, 101-136, (1979) · Zbl 1364.65223
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