A review of numerical asymptotic averaging for weakly nonlinear hyperbolic waves

    A. Krylovas Info
    R. Čiegis Info

Abstract

We present an overview of averaging method for solving weakly nonlinear hyperbolic systems. An asymptotic solution is constructed, which is uniformly valid in the “large” domain of variables t + |x| ∼ O(ϵ –1). Using this method we obtain the averaged system, which disintegrates into independent equations for the nonresonant systems. A scheme for theoretical justification of such algorithms is given and examples are presented. The averaged systems with periodic solutions are investigated for the following problems of mathematical physics: shallow water waves, gas dynamics and elastic waves. In the resonant case the averaged systems must be solved numerically. They are approximated by the finite difference schemes and the results of numerical experiments are presented.

Silpnai netiesinių hiperbolinių sistemų skaitinio asimptotinio vidurkinimo apžvalga

Santrauka. Darbe nagrinėjamas silpnai netiesinių hiperbolinių sistemų ilgųjų bangų asimptotinis sprendinys. Siūlomas jo konstravimo metodas, pagrįstas vidurkinimu bei dviejų mastelių principu. Užrašytos skirtuminės schemos suvidurkintų lygčių sistemoms spręsti. Ištirti trys periodinių asimptotinių sprendinių pavyzdžiai: sekliųjų vandenų modelis, dujų dinamikos lygtys bei tampriųjų bangų sąveika.

First Published Online: 14 Oct 2010

Keywords:

small parameter method, perturbations, hyperbolic systems, averaging, resonance, finite difference schemes, numerical solution, gas dynamics, shallow water, elastic waves

How to Cite

Krylovas, A., & Čiegis, R. (2004). A review of numerical asymptotic averaging for weakly nonlinear hyperbolic waves. Mathematical Modelling and Analysis, 9(3), 209-222. https://doi.org/10.3846/13926292.2004.9637254

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September 30, 2004
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2004-09-30

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Review Papers

How to Cite

Krylovas, A., & Čiegis, R. (2004). A review of numerical asymptotic averaging for weakly nonlinear hyperbolic waves. Mathematical Modelling and Analysis, 9(3), 209-222. https://doi.org/10.3846/13926292.2004.9637254

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