Temporal error analysis of an unconditionally energy stable second-order BDF scheme for the square phase-field crystal model

Abstract

In this paper, we first propose and study the second-order time-discrete numerical scheme for the sixth-order nonlinear parabolic problem of the square phase-field crystal model. Then, we demonstrate the two-step backward differentiation formula (BDF-2) scheme with mass conservation and energy dissipation, where the higher order nonlinear term is treated implicitly. Moreover, a rigorous error analysis is presented and we prove the optimal second-order convergence rate $O\left({\tau }^2\right)$ in $H^1$- norm, where τ is the time step. Finally, some numerical results are provided to confirm our theoretical analysis.