On the Convergence of an IEQ-based first-order Numerical Scheme for the Beris-Edwards System

TL;DR

This paper proves the convergence and stability of an energy-stable, linearly implicit numerical scheme for the Beris-Edwards liquid crystal system, using the Invariant Energy Quadratization Method, and establishes its equivalence to the original model.

Contribution

It introduces a convergent, energy-stable semi-discrete scheme for the Beris-Edwards system using IEQ, and proves its equivalence to the original system in the weak sense.

Findings

The scheme is unconditionally energy-stable.

The scheme converges to a weak solution of the system.

The reformulated system is equivalent to the original in the weak sense.

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.