This paper is focused on the numerical approximations for the hydrodynamic model of nematic liquid crystals. Under the framework of a splitting projection method, we propose a novel interior penalty discontinuous Galerkin (DG) method for solving the coupled system, which is employed by combining the scalar auxiliary variables (SAV) approach, implicit-explicit (IMEX) treatments and a rotational pressure-correction method. One prominent feature of the developed scheme here is by introducing an additional stabilization term artificially in liquid crystal equation to balance the explicit treatment for the coupling term, so that the computations of vector field from velocity field are decoupled. Hence a linear, fully decoupled and unconditionally energy-stable DG scheme can be achieved in a fully discrete manner. We show that the resulting scheme is uniquely solvable and unconditional energy stable, and the optimal error estimates of the proposed scheme are further proved theoretically. Finally, numerical results are carried out to demonstrate the accuracy and energy stability of our scheme.

中文翻译：

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向列液晶流体动力学的新型不连续伽辽金投影方案


本文主要研究向列液晶流体动力学模型的数值近似。在分裂投影方法的框架下，我们提出了一种新颖的内罚不连续伽辽金（DG）方法来求解耦合系统，该方法结合了标量辅助变量（SAV）方法、隐式-显式（IMEX）处理和旋转压力校正方法。该方案的一个显着特点是在液晶方程中人为地引入一个附加稳定项来平衡耦合项的显式处理，从而使矢量场与速度场的计算解耦。因此，可以以完全离散的方式实现线性、完全解耦和无条件能量稳定的DG方案。我们证明了所得到的方案是唯一可解且无条件能量稳定的，并且所提出方案的最优误差估计在理论上得到了进一步证明。最后，数值结果证明了我们方案的准确性和能量稳定性。