Efficient and stable time integration of Cahn-Hilliard equations: explicit, implicit and explicit iterative schemes

Abstract

To solve the Cahn-Hilliard equation numerically, a new time integration algorithm is proposed, which is based on a combination of the Eyre splitting and the local iteration modified (LIM) scheme. The latter is employed to tackle the implicit system arising each time integration step. The proposed method is gradient-stable and allows to use large time steps, whereas, regarding its computational structure, it is an explicit time integration scheme. Numerical tests are presented to demonstrate abilities of the new method and to compare it with other time integration methods for Cahn-Hilliard equation.

Figures (5)

• Figure 1: Left: an example of the "separating" free energy part $\psiup_{\text{dw}}$. Right: a typical solution and energy distribution.
• Figure 2: Convergence (achieved accuracy versus the time step size $\tauup$) for the LIM-LSS, LSS, LIM-LIE and LIE schemes on the $N=256$ space grid, $\epsilonup=\epsilonup_4(h)$
• Figure 3: Achieved accuracy versus number of matrix-vector multiplications (matvecs) for the EE and LIM schemes on the $N=256$ space grid (upper plot) and $N=512$ space grid (lower plot), $\epsilonup=\epsilonup_4(h)$. Increasing $\tauup$ in the EE scheme further is impossible due to the stability restrictions.
• Figure 4: Convergence (achieved accuracy versus the time step size $\tauup$) for the LIM-LSS, LSS, LIM-LIE and LIE schemes on the $N=256$ space grid, $\epsilonup=\epsilonup_4(1/64)$
• Figure 5: Achieved accuracy versus the number of matrix-vector multiplications (matvecs) for the EE scheme and the LIM schemes with the smoothed initial vector $\mathbf{c}^0$ and $\epsilonup=\epsilonup_4(1/64)$ on the $N=256$ grid (upper plot) and the $N=512$ grid (bottom plot). Increasing $\tauup$ in the EE scheme further is impossible due to the stability restrictions.