A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation - Guan, Zhen; Wang, Cheng; Wise, Steven M (Numerische Mathematik, 2014)
In this paper we devise a first-order-in-time, second-order-in-space, convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation. The unconditional unique solvability, energy stability and ℓ ∞ ( 0 , T ; ℓ 4 ) stability of the scheme are established. Using the a-priori stabilities, we prove error estimates for our scheme, in both the ℓ ∞ ( 0 , T ; ℓ 2 ) and ℓ ∞ ( 0 , T ; ℓ ∞ ) norms. The proofs of these estimates are notable for the fact that they do not require point-wise boundedness of the numerical solution, nor a global Lipschitz assumption or cut-off for the nonlinear term. The ℓ 2 convergence proof requires no refinement path constraint, while the one involving the ℓ ∞ norm requires only a mild linear refinement constraint, s ≤ C h . The key estimates for the error analyses take full advantage of the unconditional ℓ ∞ ( 0 , T ; ℓ 4 ) stability of the numerical solution and an interpolation estimate of the form ϕ 4 ≤ C ϕ 2 α ∇ h ϕ 2 1 - α , α = 4 - D 4 , D = 1 , 2 , 3 , which we establish for finite difference functions. We conclude the paper with some numerical tests that confirm our theoretical predictions.