Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection

Lili Ju, Xiao Li, Zhonghua Qiao, Hui Zhang
2017 Mathematics of Computation  
In this paper, we propose a class of exponential time differencing (ETD) schemes for solving the epitaxial growth model without slope selection. A linear convex splitting is first applied to the energy functional of the model, and then Fourier collocation and ETD-based multistep approximations are used respectively for spatial discretization and time integration of the corresponding gradient flow equation. Energy stabilities and error estimates of the first and second order ETD schemes are
more » ... TD schemes are rigorously established in the fully discrete sense. We also numerically demonstrate the accuracy of the proposed schemes and simulate the coarsening dynamics with small diffusion coefficients. The results show the logarithm law for the energy decay and the power laws for growth of the surface roughness and the mound width, which are consistent with the existing theories in the literature. 2010 Mathematics Subject Classification. Primary 35Q99, 47A56, 65M12, 65M70. The logarithmic term − 1 2 ln(1 + |y| 2 ), y ∈ R 2 is bounded above by zero but unbounded below. Moreover, it has no relative minima, which implies that there are no energetically favored values for |∇u|. From a physical point of view, it means that there is no slope selection mechanism in the epitaxial growth dynamics. Some detailed discussions on this issue could be found in [17, 18] and the references cited therein. The well-posedness of the initial-boundary-value problem involving the equation (1.1) was studied in [17] using the perturbation analysis method. The physically interesting process is the coarsening dynamics occurring on a very long time scale for spatially large systems, i.e., small ε. For instance, Li and Liu [18] have proved that the energy is bounded below by O(− ln t) for large time t and the global minimum energy scales as O(ln ε) in the limit ε → 0. Therefore, numerical simulations for the coarsening dynamics of large systems require the long time stability and accuracy of the numerical methods. In particular, temporally and spatially high order schemes with unconditional stability are highly demanded in term of efficiency and effectiveness. Energy stability has been investigated recently for numerical schemes of the thin film growth models [23, 24] and other phase field models [6, 10]. Wang et al. [30] derived first order (in time) convex splitting schemes for epitaxial growth models under the convex splitting framework exploited by Eyre [9], and Shen et al. [27] constructed second order (in time) schemes based on the same convex splitting approach. A linear iteration algorithm was further developed for the second order energy stable scheme for the model (1.1) in [3]. We note that these numerical schemes are nonlinear although unconditionally energy stable. A linear convex splitting scheme was developed for the model (1.1) by Chen et al. [2], and their main contribution lies in an alternate convex splitting of the Ehrlich-Schwoebel part in (1.3). The convex splitting technique also has been used extensively on different phase field models, e.g., the Cahn-Hilliard equations [20, 33], the phase field crystal model [31], the diffuse interface model with Peng-Robinson equation of state [21], etc. On the other hand, second order nonlinear and linearized Crank-Nicolson type difference schemes were derived by Qiao et al. [22] for the model (1. 1) where the unconditional energy stability is achieved with respect to a modified energy functional by introducing an auxiliary variable. For the epitaxial growth model with slope selection, Xu and Tang [32] proposed a first order linear implicit-explicit scheme by adding an order O(∆t) stabilization term of the form A∆(u n+1 − u n ), where A depends nonlinearly on the numerical solutions. In other words, it implicitly uses the L ∞ -bound assumption on |∇u n | in order to make A a controllable constant. In a recent work [19] , these technical restrictions were removed and a more reasonable stability theory was established. The linear scheme presented in [2] was essentially a first order stabilized implicit-explicit scheme with the stabilizer equal to one. The similar approaches were also applied on the Allen-Cahn and Cahn-Hilliard equations [28] . Overall, there exist very few work devoted to development of temporally high order schemes with unconditional energy stability for the model (1.1). In this paper, we will present fully discrete numerical schemes for solving the model (1.1), that uses the Fourier spectral collocation approximation for spatial discretization in combination with exponential time differencing (ETD) [1, 4, 16] and explicit multistep approximations for time integration. These schemes can be efficiently implemented via the fast Fourier transform (FFT). The ETD-based schemes often involve exact integration of the linear part of the target equation followed by an explicit approximation of the temporal integral of the nonlinear term, and can achieve high accuracy, stability and preservation of the exponential behavior of the system. Hochbruck and Ostermann provided in [13] a nice review on the exponential integrator based methods, including the ETD ones. Du and Zhu investigated the linear stabilities of some ETD schemes [7] and modified ETD schemes [8] . Ju et al. developed stable and compact ETD schemes and their fast implementations for semilinear second and fourth order parabolic equations [14, 15, 34] by utilizing suitable linear splitting techniques. However, apart from numerical implementations, theoretical analysis on stability and convergence of the ETD schemes for the phase field models are still highly desired. The rest of the paper is organized as follows. In Section 2, we first present a linear convex splitting of the energy functional (1.3), and then based on this splitting develop a class of fully discrete ETD numerical schemes, in which Fourier spectral collocation is used for spatial discretization and explicit multistep approximations for time integration. The energy stabilities of the first and second order (in time) ETD schemes are proved in Section 3, followed by error estimates rigorously derived in Section
doi:10.1090/mcom/3262 fatcat:ftwbtvffvbewzc33fxyyzo4avy