A second order linear energy stable numerical method for the Cahn–Hilliard–Hele–Shaw system

Abstract

In this paper, we consider the numerical approximations for the Cahn–Hilliard–Hele–Shaw system, which is a modified Cahn–Hilliard equation coupled with the Darcy flow law. One of challenges in solving this system numerically is how to develop linear discretization for the nonlinear term, while preserving the energy stability. By introducing a Lagrange multiplier r, we construct a fully discrete, second-order linear scheme. The proposed scheme is rigorously proven to be uniquely solvable and unconditionally stable in energy. Moreover, we show theoretical analysis on error estimates of the time step size τ and space step size h at the discrete level. Several numerical examples are presented to demonstrate the stability, accuracy and efficiency of the proposed scheme. We also show numerical simulations on the coarsening dynamics.