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An Efficient, Energy Stable Scheme for the Cahn-Hilliard-Brinkman System

Published online by Cambridge University Press:  03 June 2015

Craig Collins ,
Jie Shen  and
Steven M. Wise
Craig Collins*
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, TN 37912, USA
Jie Shen*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
Steven M. Wise*
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, TN 37912, USA
*
Email:craig.collins@math.utk.edu
Email:shen@math.purdue.edu
Corresponding author.Email:swise@math.utk.edu
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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 mod-eling 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 and 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.

Keywords

65M0665M1265M5576T99Cahn-Hilliard equationStokes equationsBrinkman equationfinite difference methodsnonlinear multigridconvex splittingenergy stability
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 4 , April 2013 , pp. 929 - 957
Copyright
Copyright © Global Science Press Limited 2013

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