Symbolic Computing and Dimensionally Reduced Models of Fluid Flow

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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. ***