Mathematical Sciences: Numerical Solution of Stiff Ordinary Differential Equations and the Navier-Stokes Equation

这方面的研究包括数值分析和流体力学的数学理论。部分工作将在有界区间上检验常微分方程的刚性系统。这些方程出现在几乎无粘流体流动问题、半导体物理和反应扩散现象的研究中。求解这些非线性方程的数值方法将在延拓和线性化的基础上发展。二维空间中不可压缩的Navier-Stokes方程也将被考虑。结果将在最小尺度的水平上进行测试。在高雷诺数的流动问题中需要这种分辨率。这项工作的结果将与均匀湍流的数值模拟直接相关。它们还将导致更好的流体流动问题算法的发展，并增加对稳定性和过渡机制的物理洞察力。

This research lies in the areas of numerical analysis and the mathematical theory of fluid dynamics. Part of the work will examine stiff systems of ordinary differential equations on a bounded interval. Such equations arise in the study of nearly inviscid fluid flow problems, semiconductor physics, and reaction-diffusion phenomena. Numerical methods to solve these nonlinear equations will be developed based on continuation and linearization. The incompressible Navier-Stokes equations in two space dimensions will also be considered. Results will be tested at the level of the smallest scales. This resolution is needed in flow problems with high Reynolds numbers. Results of this work will be immediately relevant to the numerical simulation of homogeneous turbulence. They will also lead to the development of better algorithms for fluid flow problems and increased physical insight into the mechanisms of stability and transition.