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.

翻译：本文针对基于不变能量二次化方法（IEQ）设计的流体动力学Q张量模型——即Beris-Edwards系统——提出了一种无条件能量稳定的一阶半离散数值格式的收敛性分析。该模型由描述流体流动的Navier-Stokes方程与描述液晶分子排列的Q张量梯度流耦合而成。通过采用不变能量二次化方法，我们获得了一个线性隐式格式，从而加速了计算效率。然而，该方法引入了一个辅助变量来代替体势能，且预设情况下尚不明确重构系统是否与Beris-Edwards系统等价。在本研究中，我们证明了该格式的稳定性性质，并展示了其收敛于耦合液晶系统的弱解。此外，我们还从弱意义上论证了重构系统与原系统的等价性。