Communications in Mathematical Sciences

Volume 22 (2024)

Number 1

Improved uniform error bound on the time-splitting method for the long-time dynamics of the fractional nonlinear Schrödinger equation

Pages: 1 – 14

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n1.a1

Authors

Yue Feng (Department of Mathematics, National University of Singapore; and Laboratoire Jacques-Louis Lions, Sorbonne Université, Paris, France)

Ying Ma (Department of Mathematics, Faculty of Science, Beijing University of Technology, Beijing, China)

Abstract

We establish the improved uniform error bound on the time-splitting Fourier pseudospectral (TSFP) method for the long-time dynamics of the generalized fractional nonlinear Schrödinger equation (FNLSE) with $O(\varepsilon^2)$-nonlinearity, where $\varepsilon \in (0,1]$ is a dimensionless parameter. Numerically, we discretize the FNLSE by the second-order Strang splitting method in time and Fourier pseudospectral method in space. Combining with energy method, we utilize the regularity compensation oscillation (RCO) technique to rigorously prove the improved uniform error bound at $O(h^{m_0} + \varepsilon^2 \tau^2)$ with the mesh size $h$ and time step $\tau$ up to the long-time at $O(1 / \varepsilon^2)$, which gains an additional $\varepsilon^2$ in time compared with classical error estimates. The key idea behind the RCO technique is to analyze low frequency modes by phase cancellation and control high frequency modes by the regularity of the exact solution. With the help of the RCO technique, we relax some constraints in the previous proof for the improved uniform error bound and extend the result to more general cases. Finally, numerical examples are provided to confirm our improved uniform error bound and demonstrate its suitability in different cases.

Keywords

fractional nonlinear Schrödinger equation, long-time dynamics, time-splitting Fourier pseudospectal method, improved uniform error bound, regularity compensation oscillation

2010 Mathematics Subject Classification

35Q55, 65M15, 65M70, 81Q05

Full Text (PDF format)

This work was supported by the Ministry of Education of Singapore Grant No. R-146-000-296-112 and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 850941) (Y. Feng), and the Research Foundation for Beijing University of Technology New Faculty Grant No. 006000514122521 (Y. Ma).

Received 20 April 2022

Received revised 24 April 2023

Accepted 25 April 2023

Published 7 December 2023