Volume 13, Issue 4
An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System

Craig Collins, Jie Shen & Steven M. Wise

Commun. Comput. Phys., 13 (2013), pp. 929-957.

Published online: 2013-08

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  • 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,$T$;$H^1_h$) and $ℓ^2$(0,$T$;$H^2_h$) 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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@Article{CiCP-13-929, author = {}, 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 = {

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,$T$;$H^1_h$) and $ℓ^2$(0,$T$;$H^2_h$) 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.171211.130412a}, url = {http://global-sci.org/intro/article_detail/cicp/7259.html} }
TY - JOUR T1 - An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System JO - Communications in Computational Physics VL - 4 SP - 929 EP - 957 PY - 2013 DA - 2013/08 SN - 13 DO - http://doi.org/10.4208/cicp.171211.130412a UR - https://global-sci.org/intro/article_detail/cicp/7259.html KW - AB -

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,$T$;$H^1_h$) and $ℓ^2$(0,$T$;$H^2_h$) 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.

Craig Collins, Jie Shen & Steven M. Wise. (2020). An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System. Communications in Computational Physics. 13 (4). 929-957. doi:10.4208/cicp.171211.130412a
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