130
Views
4
CrossRef citations to date
0
Altmetric
Reviews

A review of numerical asymptotic averaging for weakly nonlinear hyperbolic wavesFootnote1

&
Pages 209-222 | Received 08 Jun 2004, Published online: 14 Oct 2010

References

  • Bakhvalov , N. S. , Panasenko , G. P. and Shtaras , A. L. Springer 1999 . “ The averaging method for partial differential equations (homogenization) and its applications ” . In Partial Differential Equations V, Springer , 211 – 239 . New York
  • Chikwendu , S. C. and Kevorkian , J. 1972 . A perturbation method for hyperbolic equations with small nonlinearities . SIAMJ.Appl. Math , 22 : 235 – 258 .
  • Fornberg , B. and Driscoll , T. 1999 . A fast spectral algorithm for nonlinear wave equations with linear dispersion . J. of Comp. Phys , 155 : 456 – 467 .
  • Furihata , D. 1998 . Finite Difference Schemes that Inherit Energy Conservation or Dissipation Property 1212 Number
  • Hunter , J. K. and Keller , J. B. 1983 . Weakly nonlinear high frequency waves . Comm. Pure Appl. Math , 36 : 547 – 569 .
  • Kalyakin , L. A. 1984 . Asymptotic integration of hyperbolic equations with weakly nonlinear perturbation . Differencialnye uravnenija , 20 (2) : 351 – 353 .
  • Kalyakin , L. A. 1989 . Integrability equations as asymptotic limits of nonlinear systems . Us‐pechi Matematicheskich N auk , 44 (1) : 5 – 34 .
  • Kevorkian , J. and Cole , J. D. Springer ‐ Verlag 1996 . Multiple Scale and Singular Pertubation Methods Springer ‐ Verlag , Berlin, New‐York
  • Krylov , A. V. 1983 . About the asymptotic interaction of first order hyperbolic systems . Lith. Math. J , 23 (4) : 12 – 17 .
  • Krylov , A. V. 1986 . Asymptotic integration of weakly nonlinear partial differential systems . Zhurnal Vytchisl. Matem, i Matemat. Fiziki , 26 (1) : 72 – 79 . (in Russian)
  • Krylov , A. V. 1987 . The method of research of weakly nonlinear interaction one dimensional waves . Prikladnaja Matem, i Meck , 51 (4) : 933 – 940 . (in Russian)
  • Krylov , A. V. 1989 . The internal averaging of first order partial differential systems . Matematitcheskije Zametki , 46 (6) : 112 – 113 . (in Russian)
  • Krylov , A. V. 1989 . The substantiation of the method of the internal averaging along characteristics in weakly nonlinear systems. I . Lith. Math. J , 29 (4) : 721 – 732 .
  • Krylov , A. V. 1990 . The substantiation of the method of the internal averaging along characteristics in weakly nonlinear systems. II . Lith. Math. J , 30 (1) : 87 – 100 .
  • Krylov , A. V. 1991 . The averaging of weakly nonlinear hyperbolic systems with non‐uniform integral average . Ukrainian Math. J , 43 (5) : 611 – 618 .
  • Krylovas , A. and Čiegis , R. Asymptotic analysis of weakly nonlinear systems . Finite Difference Schemes Theory and Applications, Proceedings of the Conference FDS2000 . September 1–4 , Palanga. Edited by: A.Samarskii , A. , Čiegis , R. and Sapagovas , M. pp. 142 – 151 . Institute of Mathematics and Informatics . Vilnius
  • Krylovas , A. and Čiegis , R. 2001 . Asymptotical analysis of one dimensional gas dynamics equations . Mathematical Modelling and Analysis , 6 (1) : 103 – 112 .
  • Krylovas , A. and Čiegis , R. 2001 . Asymptotical approximation of hyperbolic weakly nonlinear systems . J. of Nonlinear Math. Phys , 8 (4) : 458 – 470 .
  • Krylovas , A. and Čiegis , R. 2003 . On the interaction of elastic waves . Journal of Civil Engineering and Management , 9 (3) : 218 – 224 .
  • Landau , L. D. and Lifshitz , E. M. 1986 . Theory of Elasticity: Course of Theoretical Physics , Oxford : Butterworth V Heinemann .
  • Majda , A. and Rosales , R. 1984 . Resonantly interacting weakly nonlinear hyperbolic waves. 1. a single space variable . Studies in Appl. Math , 71 (2) : 149 – 179 .
  • Maslov , V. P. 1987 . Asimptoticheskie metody reshenija psevdodifferencialnych uravnenii Nauka, Moskva (in Russian)
  • Maslov , V. P. and Mosolov , P. P. 1990 . Uravnenija odnomernogo barotropnogo gaza Nauka, Moskva (in Russian)
  • Nayfeh , A. H. 2000 . Pertubation Methods , New‐York : Jon Willay & Sons Inc. .
  • Samarskii , A.A. 2001 . The theory of difference schemes , New‐York, Basel : Marcel Dekker, Inc. .
  • Sanz‐Serna , J. M. 1992 . Symplectic integrators for hamiltonian problems . Acta Numerica , 1 : 243 – 286 .
  • Staras , A. L. 1977 . The asymptotic integration of weakly nonlinear partial derivatives equations . Doklady Akademii Nauk SSSR , 237 (3) : 525 – 528 .
  • Taniuti , T. 1974 . Reductive perturbation method and far fields of wave equations . Suppl. Progr. Theor. Phys , 55 : 1 – 35 .
  • This work was supported by by the Lithuanian State Science and Studies Foundation (V‐27) within the framework of the Eureka project OPTPAPER E!‐2623, E‐2002.02.27

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.