A family of second-order energy-stable schemes for Cahn–Hilliard type equations

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We focus on the numerical approximation of the Cahn–Hilliard type equations, and present a family of second-order unconditionally energy-stable schemes. By reformulating the equation into an equivalent system employing a scalar auxiliary variable, we approximate the system at the time step (n+θ) (n denoting the time step index and θ is a real-valued parameter), and devise a family of corresponding approximations that are second-order accurate and unconditionally energy stable. This family of approximations contains the often-used Crank–Nicolson scheme and the second-order backward differentiation formula as particular cases. We further develop two efficient solution algorithms for the resultant discrete system of equations to overcome the difficulty caused by the unknown scalar auxiliary variable. Within each time step, our method requires only the solution of either four de-coupled individual Helmholtz type equations, or two separate individual systems with each system consisting of two coupled Helmholtz type equations. All the resultant linear algebraic systems involve only constant and time-independent coefficient matrices that can be pre-computed. A number of numerical examples are presented to demonstrate the performance of the family of schemes developed herein. We note that this family of second-order approximations can be readily applied to devise energy-stable schemes for other types of gradient flows when combined with the auxiliary variable approaches.