ASYMPTOTIC METHOD FOR APPROXIMATION OF RESONANT INTERACTION OF NONLINEAR MULTIDIMENSIONAL HYPERBOLIC WAVES

2008 ◽  
Vol 13 (1) ◽  
pp. 47-54 ◽  
Author(s):  
A. Krylovas
Keyword(s):  
Asymptotic Method ◽  
Hyperbolic Systems ◽  
Resonant Interaction ◽  
Method Of Averaging ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Hyperbolic Waves ◽  
Wave Problems ◽  
Nonlinear Hyperbolic Systems

A method of averaging along characteristics of weakly nonlinear hyperbolic systems, which was presented in earlier works of the author for one dimensional waves, is generalized for some cases of multidimensional wave problems. In this work we consider such systems and discuss a way to use the internal averaging along characteristics for new problems of asymptotical integration.

ASYMPTOTICAL ANALYSIS OF ONE DIMENSIONAL GAS DYNAMICS EQUATIONS

2001 ◽  
Vol 6 (1) ◽  
pp. 117-128 ◽  
Author(s):  
A. Krylovas ◽  
R. Čiegis
Keyword(s):  
Gas Dynamics ◽  
Numerical Experiments ◽  
Method Of Averaging ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Resonance Case ◽  
Gas Dynamics Equations ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Asymptotical Analysis

A method of averaging is developed for constructing a uniformly valid asymptotic solution for weakly nonlinear one dimensional gas dynamics systems. Using this method we give the averaged system, which disintegrates into independent equations for the non‐resonance systems. Conditions of the resonance for periodic and almost periodic solutions are presented. In the resonance case the averaged system is solved numerically. Some results of numerical experiments are given.

scholarly journals Using the Sharp scheme of higher-order accuracy for solving some nonlinear hyperbolic systems of equations

2019 ◽  
pp. 45-53
Author(s):  
А.В. Соловьев ◽  
А.В. Данилин
Keyword(s):  
Difference Scheme ◽  
Hyperbolic Systems ◽  
Higher Order ◽  
Test Problems ◽  
Systems Of Equations ◽  
Order Accuracy ◽  
One Dimensional ◽  
Hyperbolic Systems Of Equations ◽  
Sharp Difference ◽  
Nonlinear Hyperbolic Systems

Разностная схема Диез повышенного порядка точности, ранее разработанная для решения скалярного одномерного уравнения переноса, с помощью балансно-характеристического подхода распространена на нелинейные системы уравнений мелкой воды и уравнений Эйлера. Для обеих систем уравнений решены тестовые задачи, иллюстрирующие особенности решений, полученных с помощью описываемой разностной схемы. The Sharp difference scheme of higher-order accuracy developed previously for solving the scalar one-dimensional transport equation is extended to the shallow water nonlinear systems and to the systems of Euler equations using the balance-characteristic approach. For these systems, a number of test problems are solved to illustrate the features of the solutions obtained by the described difference scheme.

A REVIEW OF NUMERICAL ASYMPTOTIC AVERAGING FOR WEAKLY NONLINEAR HYPERBOLIC WAVES

2004 ◽  
Vol 9 (3) ◽  
pp. 209-222 ◽  
Author(s):  
A. Krylovas ◽  
R. Čiegis
Keyword(s):  
Elastic Waves ◽  
Water Waves ◽  
Gas Dynamics ◽  
Averaging Method ◽  
Hyperbolic Systems ◽  
Finite Difference Schemes ◽  
Shallow Water Waves ◽  
Weakly Nonlinear ◽  
Averaged Systems ◽  
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 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.

scholarly journals Asymptotical analysis of generalized Hirota–Satsuma type system

2010 ◽  
Vol 51 ◽  
Author(s):  
Rima Kriauzienė ◽  
Aleksandras Krylovas
Keyword(s):  
Asymptotic Analysis ◽  
Nonlinear Waves ◽  
Hyperbolic Systems ◽  
Type System ◽  
Resonance Case ◽  
Weakly Nonlinear ◽  
Coupled Equations ◽  
De Vries ◽  
Asymptotical Analysis ◽  
Nonlinear Hyperbolic Systems

Paper deals with the nonlinear coupled equations of the well known in the literature Hirota–Satsuma type system. The asymptotic analysis of this system, which is based on the principle of two scales and on averaging of weakly nonlinear hyperbolic systems along characteristics is presented in the paper. The asymptotic analysis shown that the system disintegrates on three independent Korteweg–de Vries equations in the non-resonance case, and the system describes an interaction of periodical nonlinear waves in the resonance case.

Nonlinear hyperbolic waves

1988 ◽  
Vol 417 (1853) ◽  
pp. 299-308 ◽  
Keyword(s):  
Geometrical Optics ◽  
Dimensional Theory ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Hyperbolic Waves ◽  
Scale Length

A theory describing the propagation of nonlinear hyperbolic waves of any strength is developed. It is valid for small values of the wavelength, i. e. of the typical scale length of variation in the direction of propagation. At first the wave propagates along wave normals according to one-dimensional theory. It quickly splits up into a set of distinct waves, each of which soon becomes weak. The weak waves then propagate along rays according to weakly nonlinear geometrical optics.

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