We present a second-order strictly length-preserving and unconditionally energy-stable rotational discrete gradient (Rdg) scheme for the numerical approximation of the Oseen-Frank gradient flows with anisotropic elastic energy functional. Two essential ingredients of the Rdg method are reformulation of the length constrained gradient flow into an unconstrained rotational form and discrete gradient discretization for the energy variation. Besides the well-known mean-value and Gonzalez discrete gradients, we propose a novel Oseen-Frank discrete gradient, specifically designed for the solution of Oseen-Frank gradient flow. We prove that the proposed Oseen-Frank discrete gradient satisfies the energy difference relation, thus the resultant Rdg scheme is energy stable. Numerical experiments demonstrate the efficiency and accuracy of the proposed Rdg method and its capability for providing reliable simulation results with highly disparate elastic coefficients.

翻译：我们提出了一种二阶严格保长且无条件能量稳定的旋转离散梯度（Rdg）格式，用于具有各向异性弹性能量泛函的Oseen-Frank梯度流的数值逼近。Rdg方法的两个关键要素是将受长度约束的梯度流重构为无约束的旋转形式，以及采用离散梯度对能量变分进行离散化。除了众所周知的均值离散梯度和Gonzalez离散梯度外，我们还提出了一种新型Oseen-Frank离散梯度，专门针对Oseen-Frank梯度流的求解而设计。我们证明所提出的Oseen-Frank离散梯度满足能量差分关系，因此所得到的Rdg格式是能量稳定的。数值实验验证了所提出的Rdg方法的效率和准确性，以及其在弹性系数差异悬殊的情况下提供可靠模拟结果的能力。