向列相液晶流体动力学模型的高效数值方法研究

How to use minimum computational cost to get a satisfied numerical solution is always a problem that the workers in computation science attempt to solve. In particular, studying scheme with unconditional stability plays an important role in numerical algorithm design. In fact, the time step condition leads to more iterative number in time level, so it will decrease computational efficient. In this project, we focus on devising the high efficient numerical algorithm for the kinetics model of nematic liquid crystal flows. On one hand, for spatial discretization and time discretization, conforming and nonconforming finite element methods, and backward finite difference, Crank-Nicolson scheme and Crank-Nicolson Leapfrog scheme are used, respectively. One other hand, full implicit, semi-implicit, implicit/explicit schemes and extrapolation technique are applied for nonlinear terms. Further, combined with mixed finite element, characteristics finite element and operator splitting technique to research the decouple algorithms. Finally, the high efficient numerical algorithms are designed with unconditional stability. Moreover, the stability and error estimates are analyzed and the programs are designed in order to achieve numerical simulation. Then the mechanism and morphology evolution of the kinetics model of nematic liquid crystal flows can be analyzed deeply. Especially, the understand of defects of liquid crystal. Thus, the numerical analysis and algorithm design for this model are of great importance.

如何以最小的计算代价求得满足精度要求的数值解，一直以来都是科学计算工作者致力解决的问题。特别地，关于无条件稳定格式的研究，对数值算法的设计是十分重要的。事实上，对于时间步长的限制，会对时间层的迭代次数产生更多的要求，从而降低计算效率。本项目主要针对向列相液晶流体动力学模型，深入系统地研究其高效数值算法。一方面，基于协调和非协调有限元以及向后差分、Crank-Nicolson和Crank-Nicolson蛙跳格式，进行时空全离散；另一方面，基于全隐、半隐、隐显格式和外推技巧，设计非线性项的处理算法。进一步，结合混合和特征有限元方法以及算子分裂技巧，研究相应的解耦算法。最终设计出无条件稳定的高效数值算法，建立相应数值分析结果，研制可移植的计算程序，进而得以分析向列相液晶流体动力学问题的机理及其形态演化过程，尤其是加深对液晶缺陷性态的理解。该课题无论从计算理论还是算法设计上都有着十分重要的意义。

如何以最小的计算代价求得满足精度要求的数值解，一直以来都是科学计算工作者致力解决的问题。本项目主要针对向列相液晶流体动力学模型，研究其数值逼近中的高性能数值算法。针对该问题多物理场强耦合性的计算困难，我们构造了解耦的全离散格式，一方面研究了向后差分、Crank-Nicolson蛙跳等格式，另一方面基于半隐、隐显格式和外推技巧设计了非线性项的处理算法，并对于不同的物理量采用了不同时间步长。同时证明了不同时间步长下数值解的误差估计，数值模拟结果表明该方法以较少的计算时间达到与通常方法几乎一样的计算精度。进一步，为了保持该问题质量守恒性，提出了grad-div稳定化有限元方法，所构造的方法在稳定化参数增大情形下保持了计算效率和求解器稳定运行。进而建立了相应的数值分析结果，研制了可移植的计算程序。该项目在计算理论、算法设计上提供了新的研究方法。