Asymptotic approximation of hyperbolic weakly nonlinear systems

TL;DR

This paper introduces an averaging method for hyperbolic weakly nonlinear systems to obtain uniform asymptotic solutions, addressing resonance cases and demonstrating its application to shallow water equations with numerical results.

Contribution

It presents a novel averaging approach for hyperbolic systems that handles resonance conditions and provides a framework for numerical solutions in such cases.

Findings

Averaging method yields uniformly valid asymptotic solutions.

Resonance cases can be effectively handled numerically.

Application to shallow water equations demonstrates practical utility.

Abstract

An averaging method for getting uniformly valid asymptotic approximations of the solution of hyperbolic systems of equations is presented. The averaged system of equations disintegrates into independent equations for non-resonance systems. We consider the resonance conditions for some classes of solutions. The averaged system can be solved numerically in the resonance case. The shallow water problem is considered as an example of the resonance system. Results of numerical experiments are presented.