An implicit–explicit BDF2 method with variable time steps for the Ericksen–Leslie model

This work focuses on the development and analysis of a fully discrete numerical scheme for the Ericksen–Leslie model. An adjacent time step ratio $\gamma$ is defined to perform the variable time step second-order backward differentiation formulation(VBDF2). The desired discrete numerical scheme is established by combining with the standard finite element method, while the nonlinear potential is linearized using the scalar auxiliary variable(SAV) approach. To begin with, analysis of the stability and unique solvability of the VBDF2-SAV finite element scheme is shown. Secondly, the convergence rates are given by rigorous error analysis. Besides, an adaptive time-stepping strategy is designed to enhance the computational performance while maintaining accuracy. Finally, some numerical experiments are conducted to verify the analytical results and to simulate the annihilation phenomenon of singularities.