A review of numerical asymptotic averaging for weakly nonlinear hyperbolic waves
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 equ...
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| Format: | Article |
| Language: | English |
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Vilnius Gediminas Technical University
2004-09-01
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| Series: | Mathematical Modelling and Analysis |
| Subjects: | |
| Online Access: | https://journals.vgtu.lt/index.php/MMA/article/view/9730 |
| Summary: | 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
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| ISSN: | 1392-6292 1648-3510 |