Institutskolloquium

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Montag, 09.12.2019, 14:00 Uhr (WIAS-ESH)
Dr. Katharina Schratz, Heriot-Watt University Edinburgh, UK:
Nonlinear Fourier integrators for dispersive equations and beyond
mehr ... Veranstaltungsort
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Erdgeschoss, Erhard-Schmidt-Hörsaal

Abstrakt
Partial differential equations (PDEs) play a central role in mathematics, allowing us to describe physical phenomena ranging from ultra-cold atoms (Bose-Einstein condensation) up to ultra-hot matter (nuclear fusion), from learning algorithms to fluids in the human brain. To understand nature we have to understand their qualitative behavior: existence and long time behavior of solutions, their geometric and dynamical properties - as well as to compute reliably their numerical solution. While linear problems and smooth solutions are nowadays well understood, a reliable description of 'non-smooth' phenomena remains one of the most challenging open problems in computational mathematics 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. The key idea is to tackle and hardwire the underlying structure of resonances into the numerical discretization. This new approach offers strong geometric structure at low regularity and high oscillations - linking the finite dimensional discretization to powerful existence results for nonlinear PDEs at very low regularity.

 


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