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

Abstract

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