Convergence analysis of a fully discrete energy-stable numerical scheme for the Q-tensor flow of liquid crystals

TL;DR

This paper introduces a fully discrete finite difference scheme for the Q-tensor flow in liquid crystals, proving its stability and convergence, and demonstrating its effectiveness through numerical simulations.

Contribution

It provides the first convergence proof for a fully discrete scheme for Q-tensor flow, extending previous semi-discrete methods.

Findings

Scheme is stable under certain conditions

Converges to weak solutions of Q-tensor equations

Numerical simulations confirm theoretical results

Abstract

We present a fully discrete convergent finite difference scheme for the Q-tensor flow of liquid crystals based on the energy-stable semi-discrete scheme by Zhao, Yang, Gong, and Wang (Comput. Methods Appl. Mech. Engrg. 2017). We prove stability properties of the scheme and show convergence to weak solutions of the Q-tensor flow equations. We demonstrate the performance of the scheme in numerical simulations.