# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## 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 $C= C(\|u_0\|_{H^1(\mathbb {T})})>0$.
References
Similar Articles
• Retrieve articles in Mathematics of Computation with MSC (2020): 65M12, 65M15, 35Q55
• Retrieve articles in all journals with MSC (2020): 65M12, 65M15, 35Q55
• 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.