能量变分方法是复杂流体建模的常用方法之一, 该方法提供了一个直接从体系中的能量以及对应的能量关系推导出动力学微分方程的一般框架. 本学位论文中, 我们主要针对一些复杂流体的能量变分模型, 即具有动力边界条件的Cahn-Hilliard 方程、具有与反应速率相关的动力边界条件Cahn-Hilliard 方程和描述液晶分子运动的简化Ericksen-Leslie 模型, 设计出高效的、满足能量耗散关系的数值格式, 并对格式做出对应的数值分析工作. 众所周知, 能量变分模型都满足能量耗散定律, 因此设计数值格式的一个重要困难点是求解模型的数值格式也需要满足离散形式的能量耗散定律. 另外, 如何处理方程的动力边界条件、如何处理多变量的耦合以及如何让离散数值解满足约束条件等问题都是设计数值格式的困难点. 对于更有实际物理意义的具有动力边界条件的方程, 如何处理边界项也是数值分析的一个难点.

针对以上困难点, 本学位论文的主要工作分为以下三个部分：

针对具有动力边界条件的Cahn-Hilliard 方程, 我们提出了一种基于稳定因子法的数值格式. 该格式利用稳定因子项增加稳定性, 是一个关于时间一阶且满足能量递减的线性格式. 此外, 我们还给出了该格式在时间上的半离散误差估计. 最后的数值实验也验证了该格式的有效性和无条件能量稳定性等性质;

针对具有与反应速率相关的动力边界条件Cahn-Hilliard 方程, 我们提出了一个关于时间一阶的, 无条件能量稳定的线性数值格式. 我们在格式中加入稳定因子项增加稳定性. 同时, 我们给出了该时间半离散格式的误差分析. 在做误差估计时, 我们用到了迹定理来处理边界项. 最后的数值实验验证了该格式的有效性和无条件能量稳定性等性质. 此外, 我们还在数值上分别给出了当反应速率参数趋于0 和趋于无穷时的收敛结果, 这与前人的工作一致;

针对二维简化的Ericksen-Leslie 模型, 我们利用三角代换重写了该系统. 对得到的新系统, 我们提出了一个解耦的线性数值格式. 这个新的数值格式保证模长约束与无条件能量稳定性. 在特定条件下, 它满足有限元方法的离散极值原理. 因此, 在进行数值实验时, 我们保证了新系统和原系统的等价性. 最后我们给出了数值实验结果, 验证了该格式的有效性以及离散极值原理.

Energetic variational approach is commonly used to model the dynamics of complex fluids. The approach provides a general framework for deriving partial differential equations directly from the energies and the corresponding energy relationships in the system. In this thesis, we develop some efficient and energy stable numerical schemes to simulate several energetic variational models of complex fluids, i.e., the Cahn-Hilliard equation with dynamic boundary conditions, the Cahn-Hilliard equation with reaction rate dependent dynamic boundary conditions and the 2D simplified Ericksen-Leslie system, and give the corresponding error analysis. It is well-known that the energetic variational models satisfy the energy dissipation law. A significant goal in the numerical simulations is to develop the numerical schemes that can preserve the energy dissipation law at the discrete level, and then deal with the dynamic boundary conditions, the coupling problem and the constraint-preserving problem. How to deal with the boundary terms is also a difficult problem in numerical analysis for equations with dynamic boundary conditions.

Three main innovations are included in this thesis.

For the Cahn-Hilliard equation with dynamic boundary conditions, we propose a first-order in time, linear and energy stable numerical scheme, which is based on the stabilized linearly implicit approach. The energy stability of the scheme is proved and the semi-discrete-in-time error estimates are carried out. Numerical experiments are performed to validate the accuracy and the unconditional energy stability of the proposed scheme;

For the Cahn-Hilliard equation with reaction rate dependent dynamic boundary conditions, based on the stabilized linearly implicit approach, a first-order in time, linear and energy stable scheme for solving this model is proposed. The corresponding semi-discretized-in-time error estimates for the scheme are also derived, where the trace theorem is applied to deal with the boundary terms. Numerical experiments are performed to validate the accuracy and the unconditional stability of the proposed scheme. Moreover, we show the convergence results for the relaxation parameter, which are consistent with the former work.

For the 2D simplified Ericksen-Leslie system, we first rewrite the system and get a new system. For the new system, we propose an easy-to-implement time discretization scheme which preserves the sphere constraint at each node, enjoys a discrete energy law, and leads to linear and decoupled elliptic equations to be solved at each time step. A discrete maximum principle of the scheme in the finite element form is also proved, which ensures the equivalence between the simplified Ericksen-Leslie system and the new system when performing numerical experiments. Some numerical simulations are performed to simulate the dynamic motion of liquid crystals and validate the unconditional stability and the discrete maximum principle of the proposed scheme.