Convergence Analysis of an Unconditionally Energy Stable Linear Crank-Nicolson Scheme for the Cahn-Hilliard Equation (Journal of Mathematical Study, 2019)
Efficient and unconditionally stable high order time marching schemes arevery important but not easy to construct for nonlinear phase dynamics. In this paper,we propose and analysis an efficient stabilized linear Crank-Nicolson scheme for theCahn-Hilliard equation with provable unconditional stability. In this scheme the nonlinearbulk force are treated explicitly with two second-order linear stabilization terms.The semi-discretized equation is a linear elliptic system with constant coefficients, thusrobust and efficient solution procedures are guaranteed. Rigorous error analysis showthat, when the time step-size is small enough, the scheme is second order accurate intime with a prefactor controlled by some lower degree polynomial of $1/ ε.$ Here $ε$ is theinterface thickness parameter. Numerical results are presented to verify the accuracyand efficiency of the scheme.
Keywords: phase field model; Cahn-Hilliard equation; unconditionally stable; stabilized semi-implicit scheme; high order time marching.