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arXiv · 2108.04794

A fully discrete low-regularity integrator for the nonlinear Schr\"odinger equation

Alexander Ostermann·Fangyan Yao
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Abstract

For the solution of the cubic nonlinear Schr\"odinger equation in one space dimension, we propose and analyse a fully discrete low-regularity integrator. The scheme is explicit and can easily be implemented using the fast Fourier transform with a complexity of $\mathcal{O}(N\log N)$ operations per time step, where $N$ denotes the degrees of freedom in the spatial discretisation. We prove that the new scheme provides an $\mathcal{O}(\tau^{\frac32\gamma-\frac12-\varepsilon}+N^{-\gamma})$ error bound in $L^2$ for any initial data belonging to $H^\gamma$, $\frac12<\gamma\leq 1$, where $\tau$ denotes the temporal step size. Numerical examples illustrate this convergence behavior.

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Alexander Ostermann, Fangyan Yao. 2021-08-10. A fully discrete low-regularity integrator for the nonlinear Schr\"odinger equation. https://arxiv.org/abs/2108.04794

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