Highly oscillatory PDEs, slow manifolds and regularized PDE formations
Seminar Room 1, Newton Institute
The main motivation of my talk is provided by geophysical fluid dynamics. The underlying Euler or Navier-Stokes equations display oscillatory wave dynamics on a wide range of temporal and spatial scales. Simplified models are often based on the idea of balance and the concept of a slow manifold. Examples are provided by hydrostatic and geostrophic balance. One would also like to exploit these concepts on a computational level. However, slow manifolds are idealized objects that do not fully characterize the complex fluid behavior. I will describe a novel regularization technique that makes use of balance and slow manifolds in an adaptive manner. The regularization approach is based on a reinterpretation of the (linearly) implicit midpoint rule as an explicit time-stepping method applied to a regularized set of Euler equations. Adaptivity can be achieved by means of a predictor-corrector interpretation of the regularization.
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