Convergence Analysis of an Unconditionally Energy Stable Linear Crank-Nicolson Scheme for the Cahn-Hilliard Equation

摘要

Efficient and unconditionally stable high order time marching schemes arevery important but not easy to construct for nonlinear phase dynamics. In thispaper, we propose and analysis an efficient stabilized linear Crank-Nicolsonscheme for the Cahn-Hilliard equation with provable unconditional stability. Inthis scheme the nonlinear bulk force are treated explicitly with twosecond-order linear stabilization terms. The semi-discretized equation is alinear elliptic system with constant coefficients, thus robust and efficientsolution procedures are guaranteed. Rigorous error analysis show that, when thetime step-size is small enough, the scheme is second order accurate in timewith aprefactor controlled by some lower degree polynomial of $1/arepsilon$.Here $arepsilon$ is the interface thickness parameter. Numerical results arepresented to verify the accuracy and efficiency of the scheme.