Russo, Giovanni Generalized wavefront expansion properties and limitations. (English) Zbl 0609.76067 Meccanica 21, 191-199 (1986). The properties of an asymptotic method for the study of the propagation of weak shock waves in hyperbolic systems are examined in detail in the simple case of the single nonlinear wave equation in one space dimension. The general features of the method are briefly recalled and a comparison between the results obtained by this method and the exact solutions obtained by the shockfitting technique is made. Under appropriate regularity assumptions the method provides an approximation of the exact solution of the desired order for short times. Cited in 1 Document MSC: 76L05 Shock waves and blast waves in fluid mechanics 35L67 Shocks and singularities for hyperbolic equations 76M99 Basic methods in fluid mechanics 76N15 Gas dynamics (general theory) 35L65 Hyperbolic conservation laws Keywords:asymptotic method; propagation of weak shock waves; hyperbolic systems; nonlinear wave equation; one space dimension; exact solutions; shockfitting technique; regularity assumptions; approximation of the exact solution PDFBibTeX XMLCite \textit{G. Russo}, Meccanica 21, 191--199 (1986; Zbl 0609.76067) Full Text: DOI References: [1] Jeffrey A.,Quasilinear hyperbolic systems and waves, Pitman, London, 1976. · Zbl 0322.35060 [2] Rodzdestvenskii B.L.,Janenco N.N.,Systems of Quasilinear Equations and their Application to Gas Dynamics, Translations of Mathematical Monographs, vol. 55, American Mathematical Society. [3] Anile A.M.,Propagation of weak shock waves, Wave Motion, 6 (1984), 571–578, North Holland. · Zbl 0548.73012 · doi:10.1016/0165-2125(84)90047-7 [4] Anile A.M. andRusso G.,A geometric theory for the propagation of weak shock waves, Proc. of the German-Italian Symposium onApplication of Mathematics in technology, Rome, March 26–30, p. 96-123, B.G. Teubner, Stuttgart. [5] Choquet-Bruhat Y.,Ondes asymptotique et approchees pour systemes d’equations aux derivees partielles nonlineares, J. Math. Pures et Appl.,48, 1969, pp. 117–158. · Zbl 0177.36404 [6] Boillat G.,La propagation des ondes, Gauthier-Villars, Paris, 1965. · Zbl 0151.45104 [7] Landau L.D.,Lifshitz E.M., Fluid Mechanics, Pergamon Press, 1959. [8] Whitham G.B.,Linear and non linear waves, J. Wiley & Sons, New York, 1974. · Zbl 0373.76001 [9] Hunter J.K.,Keller G.B.,Weakly non linear high frequency waves, Comm. on Pure and Appl. Math., vol. XXXVI, 547–569 (1983), J. Wiley & Sons. · Zbl 0547.35070 [10] Jeffrey A.,Kawahara T.,Asymptotic Methods in Nonlinear Wave Theory, Pitman, 1982. · Zbl 0473.35002 [11] Richtmyer R.D.,Principles of Advanced Mathematical Physics, vol. I, Springer-Verlag, New York, 1978. · Zbl 0402.46001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.