Efficient local energy dissipation preserving algorithms for the Cahn–Hilliard equation

Abstract

In this paper, we show that the Cahn–Hilliard equation possesses a local energy dissipation law, which is independent of boundary conditions and produces much more information of the original problem. To inherit the intrinsic property, we derive three novel local structure-preserving algorithms for the 2D Cahn–Hilliard equation by the concatenating method. In particular, when the nonlinear bulk potential f(ϕ) in the equation is chosen as the Ginzburg–Landau double-well potential, the method discussed by Zhang and Qiao (2012) [50] is a special case of our scheme II. Thanks to the Leibnitz rules and properties of operators, the three schemes are rigorously proven to conserve the discrete local energy dissipation law in any local time–space region. Under periodic boundary conditions, the schemes are proven to possess the discrete mass conservation and total energy dissipation laws. Numerical experiments are conducted to show the performance of the proposed schemes.