In this paper, we propose two efficient fully-discrete schemes for Q-tensor flow of liquid crystals by using the first- and second-order stabilized exponential scalar auxiliary variable (sESAV) approach in time and the finite difference method for spatial discretization. The modified discrete energy dissipation laws are unconditionally satisfied for both two constructed schemes. A particular feature is that, for two-dimensional (2D) and a kind of three-dimensional (3D) Q-tensor flows, the unconditional maximum-bound-principle (MBP) preservation of the constructed first-order scheme is successfully established, and the proposed second-order scheme preserves the discrete MBP property with a mild restriction on the time-step sizes. Furthermore, we rigorously derive the corresponding error estimates for the fully-discrete second-order schemes by using the built-in stability results. Finally, various numerical examples validating the theoretical results, such as the orientation of liquid crystal in 2D and 3D, are presented for the constructed schemes.

翻译：本文针对液晶Q张量流，采用时间上的一阶和二阶稳定指数标量辅助变量（sESAV）方法结合空间离散的有限差分法，提出了两种高效的全离散格式。所构造的两种格式均无条件满足修正的离散能量耗散律。一个显著特点是，对于二维（2D）及一类三维（3D）Q张量流，成功建立了所构造一阶格式的无条件最大界原理（MBP）保持性，而所提出的二阶格式在时间步长满足温和限制条件下保持离散MBP性质。此外，我们利用内置稳定性结果严格推导了全离散二阶格式的相应误差估计。最后，通过多种数值算例（如二维和三维液晶取向）验证了所构造格式的理论结果。