基于降基法和POD方法及时空有限元法的高精度降维数值方法及应用研究

In this projection, we devote to establishing some high accuracy reduced-order numerical methods based on reduced-basis (RB) method and proper orthogonal decomposition (POD) method as well as the space-time finite element (FE) method. And at the same time, we apply them to finding the numerical solutions for non-stationary partial differential equations (PDEs). The basic idea of the project is as follows: the optimal parameters and coefficients for PDEs are determined by means of RB method, the classical space-time FE models for PDEs with the optimal parameters and coefficients are established by means of high accuracy space-time FE method, some high accuracy sample solutions are obtained on the first very short time span, snapshots are extracted from the sample solutions, the space-time POD bases are constructed from the given high accuracy snapshots by means of the POD method, the low dimensional space-time approximate spaces and reduced-order models are constructed, the space-time POD basis are renewed and developed by means of error analysis in the process of computation, the optimal numerical solutions are obtained by means of extrapolating iteration method. The method established here can simultaneously include the advantages of these three methods: preserving the relevant physical characteristics and efficiency of computation in RB method, high accuracy in space and time variables of the space-time FE method, greatly lessening degrees of freedom and computational load in POD method. When the methods are used to solve non-stationary PDE(s), they can not only keep the high accuracy, but can also reduce the degrees of freedom greatly, save computational time, improve computational efficiency, and alleviate the truncation error accumulation. The method constructed in this project is a new kind of high accuracy, high efficiency, and reduced-order method. The research for the projection dose not only hold importance theoretical value, but has also practical value in real-life applications.

项目主要建立基于降基(RB)法和特征投影分解(POD)法及时空有限元法的高精度降维数值方法，并用于求解非定常偏微分方程数值解。项目基本思想是：对带有参数和系数的偏微分方程用RB法确定参数和系数的最优值，利用高精度时空有限元法建立带有最优参数和系数的经典时空有限元模型，算出最初短时段上的样本解，提取瞬像，通过POD法从样本解构造时空POD基，建立低维时空近似解空间和降维模型，在计算过程中通过误差分析更新时空POD基，并通过外推迭代求出最优数值解。项目所研究方法同时具有RB方法保持相关物理特性和计算有效性、时空有限元法的时空高精度、POD法极大减少自由度和计算量的优点。利用该方法求解非定常偏微分方程数值解时，即能保证近似解的高精度，又能极大地减少自由度和计算量，节省计算时间，提高计算效率，减缓计算过程中的截断误差积累，是一种新型高精度高效降维方法。因此该项研究具有重要的理论价值和实际应用前景。

本项目将缩减基（RB）方法、特征投影分解（POD）技术和外推思想，引入到时空有限元方法等传统数值方法中，构造新型高精度降维时空有限元方法和降阶有限差分、有限体积等数值格式，并用于求解非定常偏微分方程数值解。项目针对时间分数阶Tricomi型方程、非线性薛定谔方程、对流扩散方程、Sobolev方程、抛物方程等分别建立了基于POD分解技术的降维有限元方法、降阶差分算法、基于RB技术的 Crank-Nicolson有限元离散格式、POD降阶有限体积外推算法、降阶谱-有限差分格式等。根据前面很短时间内的计算数据构造POD基，利用POD基的线性组合近似未知量，得到低维高效的降阶格式，该类格式在保证足够高精度、收敛性和提高收敛速度的情况下，使计算量减少到最小。证明降阶数值解和传统方法得到的数值解之间的误差，该误差决定了POD基更新准则；定义输出函数，证明后验误差估计，而后验误差估计是控制RB型方法稳定的有效手段。 . 项目组同时研究了时间间断空间连续、时空都连续的时空有限元方法，有限体积元法、扩展混合有限元法、双层网格法等，用于求解积分-微分方程、Sobolev方程、对称正则长波方程、时间Caputo型分数阶反应扩散方程、Cable方程、Burgers方程等不同方程。在时空有限元方法中，引进了有限差分方法和有限元相结合的证明方法，时间积分利用Gauss-Legendre或Gauss-Lobatto积分公式，证明了数值解的存在唯一性、稳定性、收敛性等结果，并利用数值算例验证了方法的有效性，尤其验证了时间高精度收敛性。发表标注该基金资助的论文44篇，出版学术专著1部。