Stabilization parameter analysis of a second-order linear numerical scheme for the nonlocal Cahn–Hilliard equation

Abstract

AbstractA second-order accurate (in time) and linear numerical scheme is proposed and analyzed for the nonlocal Cahn–Hilliard equation. The backward differentiation formula is used as the temporal discretization, while an explicit extrapolation is applied to the nonlinear term and the concave expansive term. In addition, an $O (\varDelta {t}^2)$ artificial regularization term, in the form of $A \varDelta _N (\phi ^{n+1} - 2 \phi ^n + \phi ^{n-1})$, is added for the sake of numerical stability. The resulting constant-coefficient linear scheme brings great numerical convenience; however, its theoretical analysis turns out to be very challenging, due to the lack of higher-order diffusion in the nonlocal model. In fact, a rough energy stability analysis can be derived, where an assumption on the $\ell ^\infty $ bound of the numerical solution is required. To recover such an $\ell ^\infty $ bound, an optimal rate convergence analysis has to be conducted, which combines a high-order consistency analysis for the numerical system and the stability estimate for the error function. We adopt a novel test function for the error equation, so that a higher-order temporal truncation error is derived to match the accuracy for discretizing the temporal derivative. Under the view that the numerical solution is actually a small perturbation of the exact solution, a uniform $\ell ^\infty $ bound of the numerical solution can be obtained, by resorting to the error estimate under a moderate constraint of the time step size. Therefore, the result of the energy stability is restated with a new assumption on the stabilization parameter $A$. Some numerical experiments are carried out to display the behavior of the proposed second-order scheme, including the convergence tests and long-time coarsening dynamics.