Abstract

In highly oscillatory fluid dynamics, such as fluids that govern Earth’s atmosphere and ocean, the role of the waves on developing and sustaining a mean, low-frequency flow is an interesting question with far-reaching consequences for how we perform numerical computations. In this talk I will discuss the role of finite-epsilon effects from time-scale separated PDEs on the formation and persistence of low-frequency phenomenon and its impact on numerical time stepping methods of the future. I will rely on the use of the semi-group operator as a mapping that helps reveal the mathematical structure of the PDE , and averaging, from common techniques used in the study of fast singular limits.   I will give examples of ODEs, and numerical computations of the shallow water equations. Finally, I will give a friendly introduction to time-parallel time-stepping methods and sketch a new proof of convergence for the parareal method when there is finite epsilon time-scale separation.