Stability analysis of several time discrete schemes for Allen–Cahn and Cahn–Hilliard equations

In this paper, the stability of several time discrete schemes for Allen–Cahn and Cahn–Hilliard equations and an error estimate for Cahn–Hilliard equation are analyzed. In order to discuss the Allen–Cahn and Cahn–Hilliard equations, a skew symmetric positive operator $\varphi$ is defined, where $\varphi=-1$ in Allen–Cahn equation and $\varphi=\Delta$ in Cahn–Hilliard equation. We analyze stabilities of some schemes for the Allen–Cahn and the Cahn–Hilliard equation. The error estimates of Cahn–Hilliard equation are based on fully discrete scheme, its main idea is to use the finite element method to discretize in space, and then use two approximate results of the elliptic projection operator to analyze. Finally, we numerically verify convergence rates of this scheme.