Effective Models and Numerical Homogenization for Wave Propagation in Heterogeneous Media on Arbitrary Timescales

Abstract

AbstractA family of effective equations for wave propagation in periodic media for arbitrary timescales $$\mathcal {O}(\varepsilon ^{-\alpha })$$ O ( ε - α ) , where $$\varepsilon \ll 1$$ ε ≪ 1 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 (Benoit and Gloria in Long-time homogenization and asymptotic ballistic transport of classical waves, 2017, arXiv:1701.08600; Allaire et al. in Crime pays; homogenized wave equations for long times, 2018, arXiv:1803.09455). The effective solutions in the family are proved to be $$\varepsilon $$ ε close to the original wave in a norm equivalent to the $${\mathrm {L}^{\infty }}(0,\varepsilon ^{-\alpha }T;{{\mathrm {L}^{2}}(\varOmega )})$$ L ∞ ( 0 , ε - α T ; L 2 ( Ω ) ) 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.