Uniform error bound of an exponential wave integrator for the long-time dynamics of the nonlinear Schr\"odinger equation with wave operator

TL;DR

This paper proves a uniform error bound for an exponential wave integrator Fourier pseudospectral method applied to the long-time dynamics of the nonlinear Schrödinger equation with wave operator, including rigorous analysis and numerical validation.

Contribution

The paper establishes the first rigorous uniform error bounds for the EWI-FP method applied to NLSW over long times, accounting for small nonlinearity parameters.

Findings

Error bound of O(h^{m-1} + ε^{2p-β} τ^2) up to time O(1/ε^β)

Numerical results confirm the sharpness of the convergence rate

Method effectively captures long-time dynamics of NLSW with rigorous error control.

Abstract

We establish the uniform error bound of an exponential wave integrator Fourier pseudospectral (EWI-FP) method for the long-time dynamics of the nonlinear Schr\"odinger equation with wave operator (NLSW), in which the strength of the nonlinearity is characterized by $\varepsilon^{2p}$ with $\varepsilon \in (0, 1]$ a dimensionless parameter and $p \in \mathbb{N}^+$. When $0 < \varepsilon \ll 1$, the long-time dynamics of the problem is equivalent to that of the NLSW with $O(1)$-nonlinearity and $O(\varepsilon)$-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 $O(h^{m-1}+\varepsilon^{2p-\beta}\tau^2)$ up to the time at $O(1/\varepsilon^{\beta})$ with $0 \leq \beta \leq 2p$, the mesh size $h$, time step $\tau$ and $m \geq 2$ 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.