@Article{EAJAM-13-4,
author = {Yue, Feng and Yichen, Guo and Yuan, Yongjun},
title = {Uniform Error Bound of an Exponential Wave Integrator for Long-Time Dynamics of the Nonlinear Schrödinger Equation with Wave Operator},
journal = {East Asian Journal on Applied Mathematics},
year = {2023},
volume = {13},
number = {4},
pages = {980--1003},
abstract = {<p style="text-align: justify;">We establish a uniform error bound of an exponential wave integrator Fourier pseudospectral (EWI-FP) method for the long-time dynamics of the nonlinear Schrödinger equation with wave operator (NLSW), in which the strength of the nonlinearity is characterized by $\varepsilon^{2p}$ with $\varepsilon\in(0,1]$&nbsp;a dimensionless parameter and $p\in \mathbb{N}^+$.&nbsp;When $0&lt;\varepsilon\ll 1,$&nbsp;the long-time dynamics of the problem is equivalent to that of the NLSW with $\mathscr{O}(1)$-nonlinearity and $\mathscr{O}(\varepsilon)$-initial data. The NLSW is numerically solved by the EWI-FP method which combines an exponential wave integrator for temporal discretization with the Fourier pseudospectral method in space. We rigorously establish the uniform $H^1$-error bound of the EWI-FP method at $\mathscr{O}(h^{m-1}+\varepsilon^{2p-\beta}\tau^2)$&nbsp;up to the time at $\mathscr{O}(1/\varepsilon^\beta)$ with $0\le \beta \le 2p$,&nbsp;the mesh size $h,$ time step $\tau$ and $m ≥ 2$ an integer depending on the regularity of the exact solution. Finally, numerical results are provided to confirm our error estimates of the EWI-FP method and show that the convergence rate is sharp.</p>},
issn = {2079-7370},
doi = {https://doi.org/10.4208/eajam.2023-100.060523},
url = {https://global-sci.com/article/82448/uniform-error-bound-of-an-exponential-wave-integrator-for-long-time-dynamics-of-the-nonlinear-schrodinger-equation-with-wave-operator}
}