An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System
Year: 2013
Author: Craig Collins, Jie Shen, Steven M. Wise
Communications in Computational Physics, Vol. 13 (2013), Iss. 4 : pp. 929–957
Abstract
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 ℓ∞(0,T;H1h) and ℓ2(0,T;H2h) 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.171211.130412a
Communications in Computational Physics, Vol. 13 (2013), Iss. 4 : pp. 929–957
Published online: 2013-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 29
Author Details
Craig Collins Email
Jie Shen Email
Steven M. Wise Email
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