# Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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## A first-order Fourier integrator for the nonlinear Schrödinger equation on $\mathbb {T}$ without loss of regularityHTML articles powered by AMS MathViewer

by Yifei Wu and Fangyan Yao
Math. Comp. 91 (2022), 1213-1235 Request permission

## Abstract:

In this paper, we propose a first-order Fourier integrator for solving the cubic nonlinear Schrödinger equation in one dimension. The scheme is explicit and can be implemented using the fast Fourier transform. By a rigorous analysis, we prove that the new scheme provides the first-order accuracy in $H^\gamma$ for any initial data belonging to $H^\gamma$, for any $\gamma >\frac 32$. That is, up to some fixed time $T$, there exists some constant $C=C(\|u\|_{L^\infty ([0,T]; H^{\gamma })})>0$, such that \begin{equation*} \|u^n-u(t_n)\|_{H^\gamma (\mathbb {T})}\le C \tau , \end{equation*} where $u^n$ denotes the numerical solution at $t_n=n\tau$. Moreover, the mass of the numerical solution $M(u^n)$ verifies \begin{equation*} \left |M(u^n)-M(u_0)\right |\le C\tau ^5. \end{equation*} In particular, our scheme does not cost any additional derivative for the first-order convergence and the numerical solution obeys the almost mass conservation law. Furthermore, if $u_0\in H^1(\mathbb {T})$, we rigorously prove that \begin{equation*} \|u^n-u(t_n)\|_{H^1(\mathbb {T})}\le C\tau ^{\frac 12-}, \end{equation*} where $C= C(\|u_0\|_{H^1(\mathbb {T})})>0$.
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• Retrieve articles in Mathematics of Computation with MSC (2020): 65M12, 65M15, 35Q55
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• Yifei Wu
• Affiliation: Center for Applied Mathematics, Tianjin University, 300072 Tianjin, People’s Republic of China
• Email: yerfmath@gmail.com
• Fangyan Yao
• Affiliation: School of Mathematical Sciences, South China University of Technology,Guangzhou, Guangdong 510640, People’s Republic of China
• MR Author ID: 1422543
• ORCID: 0000-0002-4988-7015
• Email: yfy1357@126.com
• Received by editor(s): October 8, 2020
• Received by editor(s) in revised form: July 14, 2021, September 16, 2021, and September 21, 2021
• Published electronically: December 14, 2021
• Additional Notes: The authors were partially supported by NSFC 12171356 and 11771325.