Homogenisation of parabolic/hyperbolic media

TL;DR

This paper investigates the homogenisation of media with rapidly oscillating coefficients that switch between parabolic and hyperbolic states, providing theoretical convergence results, a numerical method, and a computational study.

Contribution

It introduces a homogenisation framework for media alternating between parabolic and hyperbolic types, including convergence proofs and a numerical approximation approach.

Findings

Proved convergence of the homogenisation process.

Developed a numerical method for approximating the homogenised equations.

Conducted a numerical study demonstrating the effectiveness of the approach.

Abstract

We consider an evolutionary problem with rapidly oscillating coefficients. This causes the problem to change frequently between a parabolic and an hyperbolic state. We prove convergence of the homogenisation process in the unit square and present a numerical method to deal with approximations of the resulting equations. A numerical study finalises the contribution.