Stabilized second-order convex splitting schemes for Cahn-Hilliard models with application to diffuse-interface tumor-growth models

SUMMARYWe present unconditionally energy‐stable second‐order time‐accurate schemes for diffuse‐interface (phase‐field) models; in particular, we consider the Cahn–Hilliard equation and a diffuse‐interface tumor‐growth system consisting of a reactive Cahn–Hilliard equation and a reaction–diffusion equation. The schemes are of the Crank–Nicolson type with a new convex–concave splitting of the free energy and an artificial‐diffusivity stabilization. The case of nonconstant mobility is treated using extrapolation. For the tumor‐growth system, a semi‐implicit treatment of the reactive terms and additional stabilization are discussed. For suitable free energies, all schemes are linear. We present numerical examples that verify the second‐order accuracy, unconditional energy‐stability, and superiority compared with their first‐order accurate variants. Copyright © 2013 John Wiley & Sons, Ltd. New second‐order time‐accurate, unconditionally energy‐stable schemes for diffuse‐interface (phase‐ field) models are presented. The schemes employ a splitting of the free energy, artificial stabilization, and extrapolation. Numerical results are presented for the Cahn–Hilliard equation and a diffuse‐interface tumor‐growth model.