A projection-based time-splitting algorithm for approximating nematic liquid crystal flows with stretching

TL;DR

This paper introduces a novel projection-based time-splitting finite element method for simulating nematic liquid crystal flows with stretching, effectively handling the coupled PDE system and demonstrating stability through numerical experiments.

Contribution

It develops a new numerical scheme that uncouples the complex PDE system for nematic liquid crystals using finite elements and time-splitting, improving stability and efficiency.

Findings

The scheme effectively simulates nematic liquid crystal flows with stretching.

Numerical experiments confirm the stability and efficiency of the method.

The method handles singularity annihilation scenarios successfully.

Abstract

A numerical method is developed for solving a system of partial differential equations modeling the flow of a nematic liquid crystal fluid with stretching effect, which takes into account the geometrical shape of its molecules. This system couples the velocity vector, the scalar pressure and the director vector representing the direction along which the molecules are oriented. The scheme is designed by using finite elements in space and a time-splitting algorithm to uncouple the calculation of the variables: the velocity and pressure are computed by using a projection-based algorithm and the director is computed jointly to an auxiliary variable. Moreover, the computation of this auxiliary variable can be avoided at the discrete level by using piecewise constant finite elements in its approximation. Finally, we use a pressure stabilization technique allowing a stable equal-order interpolation for the velocity and the pressure. Numerical experiments concerning annihilation of singularities are presented to show the stability and efficiency of the scheme.