In this paper, we present a novel second-order generalised rotational discrete gradient scheme for numerically approximating the orthonormal frame gradient flow of biaxial nematic liquid crystals. This scheme relies on reformulating the original gradient flow system into an equivalent generalised "rotational" form. A second-order discrete gradient approximation of the energy variation is then devised such that it satisfies an energy difference relation. The proposed numerical scheme has two remarkable properties: (i) it strictly obeys the orthonormal property of the tensor field and (ii) it satisfies the energy dissipation law at the discrete level, regardless of the time step sizes. We provide ample numerical results to validate the accuracy, efficiency, unconditional stability and SO(3)-preserving property of this scheme. In addition, comparisons of the simulation results between the biaxial orthonormal frame gradient flow model and uniaxial Oseen-Frank gradient flow are made to demonstrate the ability of the former to characterize non-axisymmetric local anisotropy.

翻译：本文提出了一种新的二阶广义旋转离散梯度格式，用于数值逼近双轴向列液晶的正交标架梯度流。该方案基于将原始梯度流系统重新表述为等价的广义“旋转”形式。通过构建能量变化的二阶离散梯度近似，使其满足能量差分关系。所提出的数值格式具有两个显著特性：(i) 严格保持张量场的正交性；(ii) 在离散层面上满足能量耗散律，且该特性不依赖于时间步长大小。我们提供了丰富的数值结果来验证该格式的精度、效率、无条件稳定性和SO(3)保持性质。此外，通过双轴正交标架梯度流模型与单轴Oseen-Frank梯度流的模拟结果对比，展示了前者描述非轴对称局部各向异性的能力。