@Article{CiCP-13-929,
author = {Craig Collins, Jie Shen and Steven M. Wise},
title = {An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System},
journal = {Communications in Computational Physics},
year = {2013},
volume = {13},
number = {4},
pages = {929--957},
abstract = {<p style="text-align: justify;">We present an unconditionally energy stable and uniquely solvable finite
difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised
of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes model
and describes two phase very viscous flows in porous media. The scheme is based on
a convex splitting of the discrete CH energy and is semi-implicit. The equations at the
implicit time level are nonlinear, but we prove that they represent the gradient of a
strictly convex functional and are therefore uniquely solvable, regardless of time step
size. Owing to energy stability, we show that the scheme is stable in the time and space
discrete&nbsp;$ℓ^∞$(0<em>,</em>$T$;$H^1_h$) and $ℓ^2$(0<em>,</em>$T$;$H^2_h$<em>) </em>norms. We also present an efficient, practical nonlinear multigrid method – comprised of a standard FAS method for the Cahn-Hilliard
part, and a method based on the Vanka smoothing strategy for the Brinkman part – for
solving these equations. In particular, we provide evidence that the solver has nearly
optimal complexity in typical situations. The solver is applied to simulate spinodal
decomposition of a viscous fluid in a porous medium, as well as to the more general
problems of buoyancy- and boundary-driven flows.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.171211.130412a},
url = {http://global-sci.org/intro/article_detail/cicp/7259.html}
}