Constraint-preserved numerical schemes with decoupling structure for the Ericksen–Leslie model with variable density

In this paper, first- and second-order numerical schemes are developed for the Ericksen–Leslie model with variable density. The influence of variable density leads to some conflicts between decoupling and unconditional energy stability. We overcome it by special discretization and extra-fractional-step method. The fully decoupled structures are obtained by using the scalar auxiliary variable(SAV) method with only one SAV. This allows only one additional ODE to be solved so that the schemes are computationally cheaper. The non-convex constraint |b|=1 is preserved by rewriting the orientation field with polar coordinates. It also improves computational efficiency because the vector function of the orientation field is replaced by scalar function. We first reformulate the Ericksen–Leslie model as an equivalent new form by introducing a SAV and polar coordinate form of the orientation field. Secondly, two linear schemes and the corresponding unconditional energy stabilities are established. Then, we show the detailed implementations for proving that the schemes are fully decoupled and uniquely solvable. Finally, the convergence rates and energy dissipations are tested by performing some numerical experiments. The evolutionary simulations are also given.