A review of numerical asymptotic averaging for weakly nonlinear hyperbolic waves¹

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

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.

Darbe nagrinejamas silpnai netiesiniu hiperboliniu sistemu ilguju bangu asimptotinis sprendinys. Si"ulomas jo konstravimo metodas, pagristas vidurkinimu bei dvieju masteliu principu. Užrašytos skirtumines schemos suvidurkintu lygčiu sistemoms spresti. Ištirti trys periodiniu asimptotiniu sprendiniu pavyzdžiai: sekliuju vandenu modelis, duju dinamikos lygtys bei tampriuju bangu saveika.

Key words:

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

Notes

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