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arXiv · 1810.03593

Periodic homogenization of a pseudo-parabolic equation via a spatial-temporal decomposition

Arthur J. Vromans·Fons van de Ven·Adrian Muntean
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Abstract

Pseudo-parabolic equations have been used to model unsaturated fluid flow in porous media. In this paper it is shown how a pseudo-parabolic equation can be upscaled when using a spatio-temporal decomposition employed in the Peszyn'ska-Showalter-Yi paper [8]. The spatial-temporal decomposition transforms the pseudo-parabolic equation into a system containing an elliptic partial differential equation and a temporal ordinary differential equation. To strengthen our argument, the pseudo-parabolic equation has been given advection/convection/drift terms. The upscaling is done with the technique of periodic homogenization via two-scale convergence. The well-posedness of the extended pseudo-parabolic equation is shown as well. Moreover, we argue that under certain conditions, a non-local-in-time term arises from the elimination of an unknown.

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Arthur J. Vromans, Fons van de Ven, Adrian Muntean. 2018-10-08. Periodic homogenization of a pseudo-parabolic equation via a spatial-temporal decomposition. https://arxiv.org/abs/1810.03593

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