A Decoupled Unconditionally Stable Numerical Scheme for the Cahn-Hilliard-Hele-Shaw System


We propose a novel decoupled unconditionally stable numerical scheme for the simulation of two-phase flow in a Hele-Shaw cell which is governed by the Cahn-Hilliard-Hele-Shaw system (CHHS) with variable viscosity. The temporal discretization of the Cahn-Hilliard equation is based on a convex-splitting of the associated energy functional. Moreover, the capillary forcing term in the Darcy equation is separated from the pressure gradient at the time discrete level by using an operator-splitting strategy. Thus the computation of the nonlinear Cahn-Hilliard equation is completely decoupled from the update of pressure. Finally, a pressure-stabilization technique is used in the update of pressure so that at each time step one only needs to solve a Poisson equation with constant coefficient. We show that the scheme is unconditionally stable. Numerical results are presented to demonstrate the accuracy and efficiency of our scheme.


Mathematics and Statistics


This work was completed while the author was supported as a Research Assistant on an NSF Grant (DMS1312701). The author also acknowledges the support of NSF DMS1008852, a planning grant and a multidisciplinary support grant from the Florida State University. The author thanks Dr. X. Wang and Dr. S.M. Wise for some insights into the problem and many helpful conversations.

Keywords and Phrases

Nonlinear equations; Poisson equation; Convex-splitting; Decoupling; Hele-Shaw; Operator-splitting; Unconditional stability; Two phase flow; Cahn-Hilliard-Hele-Shaw

International Standard Serial Number (ISSN)

0885-7474; 1573-7691

Document Type

Article - Journal

Document Version


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© 2016 Springer Verlag, All rights reserved.

Publication Date

01 Mar 2016