Abstract
The primary challenge in designing a third-order energy-stable numerical method for the nonlocal Cahn-Hilliard equation is associated with handling the expansive concave term. We overcome this difficulty by combining it with the constant L-2 diffusion term under the diffusivity condition and employing an implicit treatment to ensure a theoretical stability analysis. In this paper, we propose and analyze a third-order accurate in time, linear stabilized numerical scheme for the nonlocal Cahn-Hilliard equation. A linearized backward differentiation formula (third-order backward differentiation formula) temporal discretization is applied, combined with the Fourier pseudo-spectral spatial discretization. The nonlinear term is approximated by an explicit extrapolation, and a third-order accurate Douglas-Dupont regularization term is added in the numerical system. With the help of this artificial regularization, a rough energy stability analysis is derived, in which a lower bound of the regularization parameter is required, dependent on the maximum norm bound of the numerical solution. The convergence analysis and error estimates are conducted, and an application of the inverse inequality recovers this functional bound. A novel test function for the error equation is taken in the form of a discrete temporal derivative, and such a test function plays a crucial role in the optimal convergence rate. Some numerical experiments, including both the convergence tests and simulation results of long-time coarsening process, are presented to demonstrate the robustness of the proposed third-order method.