An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System
Craig Collins 1, Jie Shen 2, Steven M. Wise 1*1 Department of Mathematics, University of Tennessee, Knoxville, TN 37912, USA.
2 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA.
Received 17 December 2011; Accepted (in revised version) 13 April 2012
Available online 21 September 2012
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 $l^\infty(0,T;H_h^1)$ and $l^2(0,T;H_h^2)$ 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.AMS subject classifications: 65M06, 65M12, 65M55, 76T99
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Key words: Cahn-Hilliard equation, Stokes equations, Brinkman equation, finite difference methods, nonlinear multigrid, convex splitting, energy stability.
Email: firstname.lastname@example.org (C. Collins), email@example.com (J. Shen), firstname.lastname@example.org (S. M. Wise)