Stability Analysis Of Large Time-stepping Methods For The Cahn-Hilliard Equation

The theory of the Cahn-Hilliard (CH) has been studied for decades [1, 2]. However, both the fourth and the nonlinear term make the CH equation stiff and difficult to solve numerically. For the spatial discretization, finite element schemes have been studied by, among others, Barrett et al. [5], Elliott et al. [6, 7] and Feng et al. [8]. There has been also finite difference approaches [9, 10, 11]. Fourier spectral methods have been used by Zhu et al. [12] and He et al. [4] for the CH equation with the periodic boundary conditions.This paper follows the work of [4] and [3]. The idea proposed by Xu and Tang for the molecular beam epitaxy simulation allows much larger time step than the classical time discretizations. In this work, we will use this trick towards CH equation. Precisely, in order to stabilize the calculations, an extra term, which is consistent with the order of the time discretization, is added to the classical approaches.For discretization in space, we will use the Fourier spectral approach. As for stability reason, the implicit treatment for the fourth order terms is employed. The stability properties of the constructed schemes are investigated using an energy estimation. It is proven that the decay of energy is preserved provided the magnitude of the added term is chosen suitably.The main contribution of our work is as follows: first we provide an alternative proof for the absolute stability of the first order scheme under assumption that the amplitude of the additional "A-term" is taken to be large enough. Second, we propose a second order scheme, differing from the one used by He et al. [4]. We are going to see that a slight change allows us to derive a weak energy inequality, similar to that for the MBE model [3]. Finally, we perform a series of numerical tests to confirm our claim.