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arXiv · 2301.10639

Low regularity error estimates for the time integration of 2D NLS

Lun Ji·Alexander Ostermann·Frédéric Rousset·Katharina Schratz
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Abstract

A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schr\"odinger equation on the two-dimensional torus $\mathbb{T}^2$. The scheme is analyzed in a framework of discrete Bourgain spaces, which allows us to consider initial data with low regularity; more precisely initial data in $H^s(\mathbb{T}^2)$ with $s>0$. In this way, the usual stability restriction to smooth Sobolev spaces with index $s>1$ is overcome. Rates of convergence of order $\tau^{s/2}$ in $L^2(\mathbb{T}^2)$ at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp.

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Lun Ji, Alexander Ostermann, Frédéric Rousset, Katharina Schratz. 2023-01-25. Low regularity error estimates for the time integration of 2D NLS. https://doi.org/10.1093/imanum%2Fdrae054

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