Resonances as a computational tool.
Prof. Dr. Katherina Schratz (KIT)
This seminar will be an online seminar. Please contact email@example.com to get the data for the VC.
Abstract: A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene such as for problems at low-regularity and high oscillations. Classical schemes fail to capture the oscillatory nature of the solution which leads to severe instabilities and loss of convergence. In this talk I present a new class of resonance based schemes. The key idea in the construction of the new schemes is to tackle and deeply embed the underlying nonlinear structure of resonances into the numerical discretization. As in the continuous case, these terms are central to structure preservation and offer the new schemes strong geometric properties at low regularity.
I will present the key idea behind resonances as a computational tool, give an outlook on their high order counterpart (via tree series) and numerical experiments for the Talbot effect (nonlinear dispersive quantization) leading to fractal pattern formation which is remarkably well preserved by the new schemes.