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...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Vilnius Gediminas Technical University
2004-09-01
|
| Series: | Mathematical Modelling and Analysis |
| Subjects: | |
| Online Access: | https://journals.vgtu.lt/index.php/MMA/article/view/9730 |
| id |
doaj-ef202e7195c54bf6bca41422c3e0bcab |
|---|---|
| record_format |
Article |
| spelling |
doaj-ef202e7195c54bf6bca41422c3e0bcab2021-07-02T12:31:28ZengVilnius Gediminas Technical UniversityMathematical Modelling and Analysis1392-62921648-35102004-09-019310.3846/13926292.2004.9637254A review of numerical asymptotic averaging for weakly nonlinear hyperbolic wavesA. Krylovas0R. Čiegis1Vilnius Gediminas Technical University , Sauletekio al. 11, Vilnius, LT‐10223, LithuaniaVilnius Gediminas Technical University , Sauletekio al. 11, Vilnius, LT‐10223, Lithuania 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 https://journals.vgtu.lt/index.php/MMA/article/view/9730small parameter methodperturbationshyperbolic systemsaveragingresonancefinite difference schemes |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
A. Krylovas R. Čiegis |
| spellingShingle |
A. Krylovas R. Čiegis A review of numerical asymptotic averaging for weakly nonlinear hyperbolic waves Mathematical Modelling and Analysis small parameter method perturbations hyperbolic systems averaging resonance finite difference schemes |
| author_facet |
A. Krylovas R. Čiegis |
| author_sort |
A. Krylovas |
| title |
A review of numerical asymptotic averaging for weakly nonlinear hyperbolic waves |
| title_short |
A review of numerical asymptotic averaging for weakly nonlinear hyperbolic waves |
| title_full |
A review of numerical asymptotic averaging for weakly nonlinear hyperbolic waves |
| title_fullStr |
A review of numerical asymptotic averaging for weakly nonlinear hyperbolic waves |
| title_full_unstemmed |
A review of numerical asymptotic averaging for weakly nonlinear hyperbolic waves |
| title_sort |
review of numerical asymptotic averaging for weakly nonlinear hyperbolic waves |
| publisher |
Vilnius Gediminas Technical University |
| series |
Mathematical Modelling and Analysis |
| issn |
1392-6292 1648-3510 |
| publishDate |
2004-09-01 |
| description |
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
|
| topic |
small parameter method perturbations hyperbolic systems averaging resonance finite difference schemes |
| url |
https://journals.vgtu.lt/index.php/MMA/article/view/9730 |
| work_keys_str_mv |
AT akrylovas areviewofnumericalasymptoticaveragingforweaklynonlinearhyperbolicwaves AT rciegis areviewofnumericalasymptoticaveragingforweaklynonlinearhyperbolicwaves AT akrylovas reviewofnumericalasymptoticaveragingforweaklynonlinearhyperbolicwaves AT rciegis reviewofnumericalasymptoticaveragingforweaklynonlinearhyperbolicwaves |
| _version_ |
1721330097075519488 |