Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	Dr. Katharina Schratz, Heriot-Watt University
Inviter:
Title:	Nonlinear Fourier integrators for dispersive equations
Time & Venue:	2019.09.12 10:20-11:05 N204
Abstract:	A large toolbox of numerical schemes for nonlinear 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 since the underlying PDEs have very complicated solutions exhibiting high oscillations and loss of regularity. This leads to huge errors, massive computational costs and ultimately provokes the failure of classical schemes. Nevertheless, non-smooth phenomena play a fundamental role in modern physical modeling (e.g., blow-up phenomena, turbulences, high frequencies, low dispersion limits, etc.) which makes it an essential task to develop suitable numerical schemes. In this talk I present a new class of? nonlinear Fourier integrators which offer strong geometric structure at low regularity and high oscillations. The key idea in the construction of the new schemes is to tackle and hardwire the underlying structure of resonances into the numerical discretization-linking the finite dimensional discretization to powerful existence results of nonlinear dispersive PDEs in low regularity spaces.?