A second order energy stable scheme for the CAHN-Hilliard-hele-shaw equations

Abstract

We present a second-order-in-time finite difference scheme for the Cahn-Hilliard-Hele-Shaw equations. This numerical method is uniquely solv- A ble and unconditionally energy stable. At each time step, this scheme leads to a system of nonlinear equations that can be effciently solved by a nonlinear multigrid solver. Owing to the energy stability, we derive an l2(0; T;H3h ) sta-bility of the numerical scheme. To overcome the difficulty associated with the convection term δ (φu), we perform an l(0; T;H1h) error estimate instead of the classical 1(0; T;l2) one to obtain the optimal rate convergence analysis. In addition, various numerical simulations are carried out, which demonstrate the accuracy and effciency of the proposed numerical scheme.