On the convergence of an IEQ-based first-order semi-discrete scheme for the Beris-Edwards system

¹ Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720, USA
² Department of Mathematical Sciences, University of Wisconsin Madison, 480 Lincoln Dr, Madison, WI 53706, USA

^* Corresponding author: yyue@math.wisc.edu; yukuny@andrew.cmu.edu

Received: 21 February 2023
Accepted: 22 August 2023

Abstract

We present a convergence analysis of an unconditionally energy-stable first-order semi-discrete numerical scheme designed for a hydrodynamic Q-tensor model, the so-called Beris-Edwards system, based on the Invariant Energy Quadratization Method (IEQ). The model consists of the Navier–Stokes equations for the fluid flow, coupled to the Q-tensor gradient flow describing the liquid crystal molecule alignment. By using the Invariant Energy Quadratization Method, we obtain a linearly implicit scheme, accelerating the computational speed. However, this introduces an auxiliary variable to replace the bulk potential energy and it is a priori unclear whether the reformulated system is equivalent to the Beris-Edward system. In this work, we prove stability properties of the scheme and show its convergence to a weak solution of the coupled liquid crystal system. We also demonstrate the equivalence of the reformulated and original systems in the weak sense.