材料科学中复杂相场模型的高效稳定数值方法

相场模型用在界面区域上连续守恒和非守恒的场变量描述材料科学中涉及许多复杂过程的微结构，它的计算是近年来发展的一种强有力的模拟和预测材料中尺度形态和微结构演化的方法。由于材料科学中复杂相场模型的耦合非线性特点，显式或简单的全隐式离散格式将会导致能量不稳定格式或计算中出现效率较低且不易求解的非线性问题。本项目旨在利用标量辅助变量方法和指数时间差分方法的思想，分别通过引入辅助变量将能量泛函分裂为二次形式和结合龙格库塔方法，对材料科学中复杂相场模型设计时间上高阶、无条件能量稳定、易于计算的解耦线性隐式数值格式。在理论上利用梯度流问题的能量耗散定律，分析半离散数值格式的稳定性及全离散数值格式的误差和收敛性；在数值计算上，利用上述稳定数值格式进而设计快速算法，验证数值格式的准确性和稳定性，并模拟和预测材料微结构的演化过程，对深刻了解材料形变机理和及时更新材料具有重要的理论价值和应用前景。

The phase field model describes a microstructure by adopting a set of conserved and nonconserved field variables that are continuous across the interfacial regions. The phase field approach has recently emerged as an important and extremely versatile technique for modeling and predicting mesoscale morphological and microstructure evolution in materials. Due to the coupled nonlinear term of the complex multi-component multi-phase field model in materials, therefore simple fully-implicit or explicit type discretizations for the phase field model will induce unfavorable nonlinear and/or energy unstable schemes which are not efficient in practice. How to develop linear and/or decoupled schemes while pre-serving the energy stability at the discrete level for solving the nonlinear coupled system numerically has become one main topic for the algorithmic design. Based on the ideas of the scalar auxiliary variable and exponential time differencing approaches, we aim to design the high order, unconditionally energy stable and efficient numerical decoupled implicit schemes for the complex phase field model in material, the main idea is to reformulate and split the energy functional in the quadratic form by introducing extra variables and combine the stabilized integrating factor with Runge-Kutta method. From the point of view of the theoretical analysis, the rigorous stability and error analysis for the schemes are analyzed by applying the energy dissipation law of the gradient flow problems. From the numerical point of view, some fast algorithms and adaptive algorithms are developed for the energy stable schemes, and a series of numerical experiments is carried out to illustrate the efficiency of the proposed efficient unconditionally stable numerical schemes, and simulate and predict the microstructure evolution. The establishment of accurate, highly efficient numerical schemes and completed numerical analysis has important theoretical value and application aspects for deeply revealing the microstructure evolution and guiding science engineering practice.