Effective equations of arbitrary order for wave propagation in periodic media

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

Assyr Abdulle, Timothée Noé Pouchon

A family of effective equations for wave propagation in periodic media for arbitrary timescales O(epsilon-alpha), where epsilon MUCH LESS-THAN1 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, ; Allaire et al. in Crime pays; homogenized wave equations for long times, 2018, ). The effective solutions in the family are proved to be epsilon close to the original wave in a norm equivalent to the L infinity(0,epsilon-alpha T; L2(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.

SPRINGER2020

Effective models for the multidimensional wave equation in heterogeneous media over long time and numerical homogenization

Assyr Abdulle, Timothée Noé Pouchon

A family of effective equations that capture the long time dispersive effects of wave propagation in heterogeneous media in an arbitrary large periodic spatial domain Omega subset of R-d is proposed and analyzed. For a wave equation with highly oscillatory periodic spatial tensors of characteristic length epsilon, we prove that the solution of any member of our family of effective equations is epsilon-close to the true oscillatory wave over a time interval of length T-epsilon = O(epsilon(-2)) in a norm equivalent to the L-infinity(0, T-epsilon; L-2(Omega)) norm. We show that the previously derived effective equation in [T. Dohnal, A. Lamacz and B. Schweizer, Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Model. Simulat. 12 (2014) 488-513] belongs to our family of effective equations. Moreover, while Bloch wave techniques were previously used, we show that asymptotic expansion techniques give an alternative way to derive such effective equations. An algorithm to compute the tensors involved in the dispersive equation and allowing for efficient numerical homogenization methods over long time is proposed.

World Scientific Publ Co Pte Ltd2016

MATHICSE Technical Report : Effective models for long time wave propagation in locally periodic media

Assyr Abdulle, Timothée Noé Pouchon

A family of effective equations for the wave equation in locally periodic media over long time is derived. In particular, explicit formulas for the effective tensors are provided. To validate the derivation, an a priori error estimate between the effective solutions and the original wave is proved. As the dependence of the estimate on the domain is explicit, the result holds in arbitrarily large periodic hypercube. This constitutes the first analysis for the description of long time effects for the wave equation in locally periodic media. Thanks to this result, the long time a priori error analysis of the numerical homogenization method presented in [A. Abdulle and T. Pouchon, SIAM J. Numer. Anal., 54, 2016, pp. 1507–1534] is generalized to the case of a locally periodic tensor.

MATHICSE2017