A Decoupled Energy Stable Numerical Scheme for the Modified Cahn–Hilliard–Hele–Shaw System with Logarithmic Potential

Abstract

A decoupled unconditionally stable numerical scheme for the modified Cahn–Hilliard–Hele–Shaw system with logarithmic potential is proposed in this paper. Based on the convex-splitting of the associated energy functional, the temporal discretization of the scheme is given. The fractional step method is used to decouple the nonlinear modified Cahn–Hilliard equation from the pressure gradient. Then, at each time step, one only needs to solve Poisson’s equation which is obtained by using an incremental pressure-stabilization technique. In terms of logarithmic potential, using the regularization procedure can make the domain extended from $\left(-\mathrm{1,1}\right)$ to $\left(-\infty ,\infty \right)$ for $F\left(\varphi \right)$ . Finally, the proof of the energy stability and the calculation of the error estimate are presented. Numerical examples are recorded to illustrate the accuracy and performance for the proposed scheme.