# Effective models and numerical homogenization for wave propagation in heterogeneous media on arbitrary timescales

## TL;DR

This paper introduces a family of effective equations for wave propagation in periodic media valid over arbitrary timescales, ensuring well-posedness without regularization and providing a computationally efficient method for calculating effective tensors.

## Contribution

The paper develops a new family of effective wave equations for arbitrary timescales, proving their well-posedness and introducing a novel, cost-effective numerical method for computing effective tensors of any order.

## Key findings

- Effective equations are well-posed without regularization.
- Effective solutions are close to original wave solutions in a specific norm.
- A new relation reduces computational cost for effective tensor calculation.

## Abstract

A family of effective equations for wave propagation in periodic media for arbitrary timescales $\mathcal{O}(\varepsilon^{-\alpha})$, where $\varepsilon\ll1$ is the period of the tensor describing the medium, is proposed. The well-posedness of the effective equations of the family is ensured without requiring a regularization process as in previous models [A. Benoit and A. Gloria, arXiv:1701.08600, 2017], [G. Allaire, A. Lamacz, and J. Rauch, arXiv:1803.09455, 2018]. The effective solutions in the family are proved to be $\varepsilon$ close to the original wave in a norm equivalent to the $L^\infty(0,\varepsilon^{-\alpha}T;L^2(\Omega))$ norm. In addition, a numerical procedure for the computation of the effective tensors of arbitrary order is provided. In particular, we present a new relation between the correctors of arbitrary order, which allows to substantially reduce the computational cost of the effective tensors of arbitrary order. This relation is not limited to the effective equations presented in this paper and can be used to compute the effective tensors of alternative effective models.