Invited Lecture
A low-regularity Fourier integrator for the cubic nonlinear Schrödinger equation
University of Innsbruck, Austria
Abstract
A new filtered low-regularity Fourier integrator for the cubic nonlinear Schrödinger equation is presented. This scheme has better convergence rates at low regularity than any known scheme in the literature so far. To prove this superior error behavior, we combine the better local error properties of the new scheme with a stability analysis based on general discrete Strichartz-type estimates. The latter allow us to handle a much rougher class of solutions as the error analysis can be carried out directly at the level of $L^2$. We are able to establish a global error estimate in $L^2$ for $H^1$ solutions, which is roughly of order $\tau^{ \frac12 + \frac{5-d}{12} }$ in dimension $d \leq 3$ with $\tau$ denoting the time step size. For details, see arxiv.org/pdf/1902.06779.pdf
This is joint work with Frédéric Rousset (Université Paris-Sud) and Katharina Schratz (KIT, Karlsruhe).
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