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We present a convergence analysis of an unconditionally energy-stablefirst-order semi-discrete numerical scheme designed for a hydrodynamic Q-tensormodel, the so-called Beris-Edwards system, based on the Invariant EnergyQuadratization Method (IEQ). The model consists of the Navier-Stokes equationsfor the fluid flow, coupled to the Q-tensor gradient flow describing the liquidcrystal molecule alignment. By using the Invariant Energy QuadratizationMethod, we obtain a linearly implicit scheme, accelerating the computationalspeed. However, this introduces an auxiliary variable to replace the bulkpotential energy and it is a priori unclear whether the reformulated system isequivalent to the Beris-Edward system. In this work, we prove stabilityproperties of the scheme and show its convergence to a weak solution of thecoupled liquid crystal system. We also demonstrate the equivalence of thereformulated and original systems in the weak sense.

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