An energy stable fourth order finite difference scheme for the Cahn–Hilliard equation

• Published in: Journal of computational and applied mathematics Vol. 362; pp. 574 - 595
• Main Authors: Cheng, Kelong, Feng, Wenqiang, Wang, Cheng, Wise, Steven M.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.12.2019
• Subjects: Cahn–Hilliard equation; Energy stability; Long stencil fourth order finite difference approximation; Optimal rate convergence analysis; Preconditioned steepest descent iteration; Second order accuracy in time; Second order accuracy in time; 65M06; Energy stability; Optimal rate convergence analysis; 65K10; Long stencil fourth order finite difference approximation; 35K35; 65M12; Preconditioned steepest descent iteration; 35K55; Cahn–Hilliard equation
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2018.05.039

In this paper we propose and analyze an energy stable numerical scheme for the Cahn–Hilliard equation, with second order accuracy in time and the fourth order finite difference approximation in space. In particular, the truncation error for the long stencil fourth order finite difference approximation, over a uniform numerical grid with a periodic boundary condition, is analyzed, via the help of discrete Fourier analysis instead of the standard Taylor expansion. This in turn results in a reduced regularity requirement for the test function. In the temporal approximation, we apply a second order BDF stencil, combined with a second order extrapolation formula applied to the concave diffusion term, as well as a second order artificial Douglas–Dupont regularization term, for the sake of energy stability. As a result, the unique solvability, energy stability are established for the proposed numerical scheme, and an optimal rate convergence analysis is derived in the ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh2) norm. A few numerical experiments are presented, which confirm the robustness and accuracy of the proposed scheme.