Analytical and numerical study of weakly nonlinear hyperbolic waves in a van der Waals gas

TL;DR

This paper investigates weakly nonlinear hyperbolic waves in a van der Waals gas using asymptotic analysis, deriving an integro-differential evolution equation, and studying traveling wave solutions both analytically and numerically.

Contribution

It introduces a novel integro-differential model for wave interactions in a van der Waals gas and analyzes traveling wave solutions with numerical validation.

Findings

Traveling wave solutions depend on van der Waals parameters.

Numerical experiments show non-breaking solutions persist over time.

The convolution term significantly influences wave behavior.

Abstract

In this paper, we characterized resonant interaction of weakly nonlinear hyperbolic waves in gas dynamics with a real gas background. An asymptotic approach is used to study the interaction between waves, governed by the Euler equations of gas dynamics, supplemented by a van der Waal equation of state; the evolution equation is an integro-differential equation composed of a Burgers type nonlinear term and a convolution term with a known kernel. A one parameter family of traveling wave solutions for various values of van der Waals parameter is studied analytically and numerically. Effect of the influence of van der Waals parameter on the properties of traveling waves solution is investigated. Numerical experiments on the evolution of arbitrary initial data are performed using fractional step algorithm, describing the behavior of convolution term in the integrodiffrential equation. The existence of non breaking for all time solutions is substantiated numerically.