Error estimates of a sphere-constraint-preserving numerical scheme for Ericksen-Leslie system with variable density

In this paper, a new first-order, sphere-constraint-preserving numerical scheme is constructed for the Ericksen-Leslie equations with variable density. Firstly, by denoting orientation field vector $ \pmb{d} in polar coordinate, we rewrite system into an equivalent such that sphere constraint \lvert \pmb{d}\rvert = 1 can be still preserved at discrete level. Secondly, propose first-order discretization time which velocity and modified pressure are determined generalized Stokes problem each step. Then, unconditional energy stability rigorous error estimates both derived. Finally, some simulations two dimensions provided to demonstrate of accuracy presented scheme.