Abstract
A family of effective equations for wave propagation in periodic media for arbitrary timescales mathcal {O}(varepsilon ^{-alpha }), where varepsilon ll 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 close to the original wave in a norm equivalent to the {mathrm {L}^{infty }}(0,varepsilon ^{-alpha }T;{{mathrm {L}^{2}}(varOmega )}) 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.
Highlights
The wave equation in heterogeneous media is widely used in many applications such as seismic inversion, medical imaging or the manufacture of composite materials
The second main result of the paper is an explicit procedure for the computation of the high order effective tensors {a2r, b2r } in (4), for which we provide a new relation between the high order correctors
We presented a family of effective equations for wave propagation in periodic media for arbitrary timescales
Summary
The wave equation in heterogeneous media is widely used in many applications such as seismic inversion, medical imaging or the manufacture of composite materials. Under assumption (3), a0 is proved to be constant and an explicit formula is obtained (see, e.g., [12,14,17,28]): it can be computed by means of the first-order correctors, which are defined as the solutions of cell problems (i.e., elliptic equations in Y based on a(y) with periodic boundary conditions). The first main result is the definition of a family of effective equations that approximate uε for arbitrary timescales O(ε−α). The second main result of the paper is an explicit procedure for the computation of the high order effective tensors {a2r , b2r } in (4), for which we provide a new relation between the high order correctors. We present our second main result: a relation between the correctors which allows to reduce the computational cost of the effective tensors. L∞(0, T ε; L02(Ω)) and ∂t2uε ∈ L2(0, T ε; Wp∗er(Ω))
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