Fully decoupled linear BDF2 scheme for the penalty incompressible Ericksen–Leslie equations

Abstract

This work focuses on the numerical approximation of the penalized Ericksen–Leslie equations. A second-order numerical scheme which has the advantages of full decoupling, linearization and unconditional stability in energy is constructed. Firstly, an equivalent new system is established by introducing two scalar auxiliary variables. One is added to the convective term in the Navier–Stokes equation and the other is added to the rest nonlinear terms. Secondly, a second-order BDF(backward differentiation formula) scheme for the new system is constructed while the pressure-correction method is used and the unconditional stability in energy is subsequently proved. Thirdly, the detailed implementation process is given to clarify that the scheme is fully decoupled and linear. Finally, several numerical simulations are shown to illustrate the accuracy and effectiveness of the scheme. Moreover, the annihilation phenomena of singularities are also performed.