Two SAV numerical methods for the nonlocal Cahn-Hilliard-Hele-Shaw system.

The capability of nonlocal models in materials field has attracted the scientific community a great attention to characterize the effect of various types of material heterogeneities and defects. In this paper, we are concerned with construction of a energy stability method for the nonlocal Cahn–Hilliard–Hele-Shaw system under periodic boundary conditions. By employing the Fourier spectral method in spatial and the scalar auxiliary variable (SAV) approach with first/second-order backward differentiation formula in temporal, two fast and effective schemes are established. The unconditional energy stability analyses is rigorously derived. Numerical experiments are presented to verify our theoretical results and to show the robustness and accuracy of the proposed method.