自然对流问题的高效数值方法研究

How to improve computational efficiency is always a problem that the workers in computation science and engineering attempt to solve. Specifically, it is hoped that using the minimum computational cost to get a satisfied solution. In this project, we focus on devising the efficiently high precision algorithm for the natural convection problem. For spatial discretization and time discretization, the stabilized conforming and nonconforming finite element methods, and the adaptive time-stepping method are used respectively. Newton iteration or other is applied for nonlinear terms. The decoupled technique is designed to reduce the computational cost for solving the discrete system. In order to improve stability and simplify complexity, based on the stabilized finite element methods, we develop an efficient two-step scheme. Besides, the long-time stability and error estimates are analyzed and the programs are designed in order to achieve numerical simulation. Moreover, the effectiveness of the given method is demonstrated by compared with some previous schemes. We will give the numerical results of the natural convection problem with large Rayleigh number to study the properties of the solutions. Then mechanism and morphology evolution of the natural convection problem can be analyzed, and the essence can be understood deeply. Furthermore, we provide new research approaches to develop the nonlinear scientific research and apply to the computational heat transfer in the engineering technology.

如何提高计算效率一直是科学与工程计算工作者致力解决的问题。具体地说，就是希望所设计的数值算法能够以最小的计算代价求得满足精度要求的数值解。本项目针对自然对流问题，研究其数值逼近中的高性能算法。空间离散运用稳定化协调和非协调有限元方法逼近，时间离散使用自适应时间步长方法，非线性项采用牛顿迭代等方法，离散系统利用解耦技巧降低求解方程组的复杂度。为提高数值计算的稳定性和简化计算的复杂性，运用稳定化方法，我们将设计出高效二步算法。此外，数值分析方面，给出算法的收敛性分析、最优误差估计和长时间稳定性分析；数值计算方面，研制友好的计算程序，实现数值模拟，并通过与已有的一些数值格式比较，来说明新方法的高效性，进而对大瑞利数自然对流问题进行模拟，研究解的性态。从而得以分析该问题发展的机理及其形态演化过程，能够更加深刻地理解它的本质，为非线性科学的研究和发展以及计算传热学在工程中的应用提供新的研究途径。

如何提高计算效率一直是科学与工程计算工作者致力解决的问题。具体地说，就是希望所设计的数值算法能够以最小的计算代价求得满足精度要求的数值解。本项目针对自然对流问题，研究了其数值逼近中的高性能算法。我们构造了定常自然对流问题高效二步算法，空间离散基于最低次等阶和二次等阶速度、压力和温度空间配对的稳定化协调有限元方法。进而研究离散格式的稳定性和收敛性，最后通过数值算例验证了算法的有效性。进一步，研究了非定常自然对流问题的特征变分多尺度有限元方法，相比于经典的Galerkin有限元方法，该方法能够节省大量的计算时间，并能有效地处理大瑞利数问题。从而得以分析该问题发展的机理及其形态演化过程，能够更加深刻地理解它的本质，为非线性科学的研究和发展以及计算传热学在工程中的应用提供新的研究途径。

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