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

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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.