  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.
