In this paper, first- and second-order numerical schemes are developed for the Ericksen–Leslie model with variable density. The influence of variable density leads to some conflicts between decoupling and unconditional energy stability. We overcome it by special discretization and extra-fractional-step method. The fully decoupled structures are obtained by using the scalar auxiliary variable (SAV) method with only one SAV. This allows only one additional ODE to be solved so that the schemes are computationally cheaper. The non-convex constraint is preserved by rewriting the orientation field with polar coordinates. It also improves computational efficiency because the vector function of the orientation field is replaced by scalar function. We first reformulate the Ericksen–Leslie model as an equivalent new form by introducing a SAV and polar coordinate form of the orientation field. Secondly, two linear schemes and the corresponding unconditional energy stabilities are established. Then, we show the detailed implementations for proving that the schemes are fully decoupled and uniquely solvable. Finally, the convergence rates and energy dissipations are tested by performing some numerical experiments. The evolutionary simulations are also given.

中文翻译：

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变密度 Ericksen-Leslie 模型具有解耦结构的保留约束数值格式


在本文中，为变密度的 Ericksen-Leslie 模型开发了一阶和二阶数值格式。变密度的影响导致解耦与无条件能量稳定之间存在一些冲突。我们通过特殊的离散化和超分数阶方法克服了它。完全解耦的结构是通过使用标量辅助变量（SAV）方法获得的，只有一个SAV。这仅允许求解一个额外的 ODE，因此该方案的计算成本更低。通过用极坐标重写方向场来保留非凸约束。由于方向场的矢量函数被标量函数代替，因此还提高了计算效率。我们首先通过引入 SAV 和方向场的极坐标形式，将 Ericksen-Leslie 模型重新表述为等效的新形式。其次，建立了两个线性格式和相应的无条件能量稳定性。然后，我们展示了详细的实现，以证明该方案是完全解耦且唯一可解的。最后，通过进行一些数值实验来测试收敛速度和能量耗散。还给出了进化模拟。