Symbolic Computing and Dimensionally Reduced Models of Fluid Flow

该项目将开发新的方法，对全球环境模拟问题中出现的那种不可压缩流体流动进行数值模拟。该模型是由Navier-Stokes方程导出的层状不可压缩流的双曲型和弥散型垂直平均模型。这些模型将三维模型简化为复杂的二维模型系统，从而保证了更高的数值效率。然而，平均模型在某些有趣的参数范围内并不是很好的适定性。因此，必须研究它们的不稳定性和正则性。这种分析以及连续介质方程的离散化、离散问题的精度和稳定性分析以及离散方程的编码都是代数复杂的，因此计算机代数的使用将在这个项目中发挥关键作用。该项目的结果可用于海洋、大气和其他复杂环境的全球模拟。目前实现这种环境的详细模型的代码需要计算资源，即使是最大的计算机也会负担得起。该项目的目标是为它们的解产生新的模型和数值方法，这将大大提高一些重要应用的现有模型的效率和精度。***

This project will develop new methods for numerically modelling incompressible fluid flows of the type that appear in global environmental modelling problems. The proposed models are hyperbolic and dispersive vertically-averaged models of stratified incompressible flow which are derived from the Navier-Stokes equations. These models reduce three-dimensional models to a system of complex two-dimensional models, which promise greater numerical efficiency. However, the averaged models are not well posed for some interesting ranges of their parameters. Consequently, their instability and regularization must be investigated. This analysis, along with the discretization of the continuum equations, the accuracy and stability analysis of the discrete problems, and the coding of the discrete equations are all algebraically complex, so the use of computer algebra will play a critical role in this project. The results of the project can be used for the global modelling of the ocean, the atmosphere, and other complex environments. The current codes implementing detailed models of such environments require computational resources that tax even the largest computers. The goal of this project is to produce new models and numerical methods for their solution which would represent a significant improvement in efficiency and accuracy over existing models for some important applications. ***