非定常流体力学方程基于特征正交分解及自适应网格加密的外推降维数值解法研究

In this projection, we intend to establish some new reduced- dimensional finite difference (FD), finite element (FE), or finite volume element (FVE) extrapolation algorithms based on proper orthogonal decomposition (POD) method and adaptive mesh refinement of posteriori error estimates and apply these new reduced-dimensional FD, FE, or FVE extrapolations algorithns based on adaptive local mesh refinement techniques with posteriori error estimates and based on adaptive POD bases updated automatically to finding the numerical solutions for non- stationary fluid mechanics equations. These new methods have the following advantages: under the conditions of ensuring sufficiently high accuracy numerical solutions, they can greatly reduce the dimensions of discrete FD, FE, and FVE formulations of fluid mechanics equations, save the memory requirements, and alleviate the computational laod such that these new methods can quickly forecast and simulate the actual fluid flow in practical engineering problems with very high computing efficiency and sufficiently high accuracy, which achieves efficiently and quickly calculating and forecasting real fluid field flow. These new FD, FE, or FVE extrapolation algorithms with double adaptive techniques would play an important in implement of simulations and forecasts in flow fluid field, become some new computational methods with high-accuracy and high-efficiency, and serve the national economic development. It follows that the projection does not only hold important theory value, but has also actual applied value.

该项目拟创建基于特征投影分解(Proper Orthogonal Decomposition,简记为POD)技术和基于后验误差估计的自适应有限差分、有限元或有限体积元外推降维新方法，将这些具有自适应网格局部加密和自适应POD基自动更新的有限差分、有限元或有限体积元外推降维新方法用于求非定常流体力学方程数值解。这些新方法优点在于：保证具有足够高精确度数值解前提下，大幅度地降低非定常流体力学方程离散格式的维数，大大减少内存要求和计算时间，使这些降维方法在实际工程问题的数值计算中，能以极高的计算效率和足够高的计算精度对实际流场流动进行预测或模拟，达到高效快速地预测和预报及模拟流场流动的目的，使这些具有双重自适应技术的有限差分、有限元或有限体元外推降维方法在流场流动模拟和预测实施中产生重大作用，成为高精度和高性能的新型计算方法，为国家经济建设服务。因此该项目既具有重要的理论价值又具有实际应用价值。

在对该项目研究中，我们开展了基于特征投影分解(Proper Orthogonal Decomposition,简记为POD)技术和基于后验误差估计的自适应有限差分、有限元或有限体积元外推降阶新方法的研究，将具有自适应网格局部加密和自适应POD基自动更新的有限差分、有限元或有限体积元外推降维新方法用于对求非定常流体力学方程数值解研究。发表了相关论文52篇(其中SCI收录34篇、EI收录1篇、北大中文核心期刊17篇)，一些结果在国际上都是我们首先提出来的。这些新方法的优点是：保证具有足够高精确度数值解前提下，大幅度地降低非定常流体力学方程离散格式的维数，大大减少内存要求和计算时间，使这些降维方法能以极高的计算效率和足够高的计算精度对实际物理现象进行预测或模拟，能高效快速地预测和预报及模拟物理规律。这些具有双重自适应技术的有限差分、有限元或有限体元外推降维方法已经成为高精度和高性能的新型计算方法，在计算数学和计算流体力学中产生了很好的效果。因此对该项目的研究既具有重要的理论价值又具有实际应用价值。

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