Numerical Study of the 2D Cahn–Hilliard Model of Phase Separation with Logarithmic Potential

In this paper, we have approximated the solutions of the Cahn–Hilliard equation (CH) with the logarithmic potential function which arises in the modeling of phase separation of binary alloys. The CH equation is a high-order nonlinear equation, consequently, utilizing a common difference scheme on the CH equation causes long stencil schemes. To resolve the faults of long stencil schemes, we split the CH equation to second-order system under the Neumann boundary condition and we applied second-order scheme based on the 2D Crank–Nicolson method to discrete it. The uniqueness and error estimation of the approximated solution is proved. Also, preserving the conservation of mass and decreasing the total energy are investigated. Finally, to confirm the theoretical results, three examples with various initial conditions are presented.