Uniform and optimal error estimates of an exponential wave integrator sine pseudospectral method for the nonlinear Schrodinger equation with wave operator

TL;DR

This paper introduces an exponential wave integrator sine pseudospectral method for the nonlinear Schrödinger equation with wave operator, providing uniform error bounds and spectral accuracy, and addressing challenges from rapid oscillations as the perturbation parameter approaches zero.

Contribution

The paper develops a new numerical method with rigorous uniform error estimates for the NLS with wave operator, improving upon existing finite difference approaches.

Findings

Achieves optimal uniform error bounds of O(τ^2) and O(τ) for well- and ill-prepared data

Provides spectral accuracy in spatial discretization in L^2 and semi-H^1 norms

Numerical results confirm theoretical error estimates

Abstract

We propose an exponential wave integrator sine pseudospectral (EWI-SP) method for the nonlinear Schr\"{o}dinger equation (NLS) with wave operator (NLSW), and carry out rigorous error analysis. The NLSW is NLS perturbed by the wave operator with strength described by a dimensionless parameter $\varepsilon\in(0,1]$. As $\varepsilon\to0^+$, the NLSW converges to the NLS and for the small perturbation, i.e. $0<\varepsilon\ll1$, the solution of the NLSW differs from that of the NLS with a function oscillating in time with $O(\varepsilon^2)$-wavelength at $O(\varepsilon^2)$ and $O(\varepsilon^4)$ amplitudes for ill-prepared and well-prepared initial data, respectively. This rapid oscillation in time brings significant difficulties in designing and analyzing numerical methods with error bounds uniformly in $\varepsilon$. In this work, we show that the proposed EWI-SP possesses the optimal uniform error bounds at $O(\tau^2)$ and $O(\tau)$ in $\tau$ (time step) for well-prepared initial data and ill-prepared initial data, respectively, and spectral accuracy in $h$ (mesh size) for the both cases, in the $L^2$ and semi-$H^1$ norms. This result significantly improves the error bounds of the finite difference methods for the NLSW. Our approach involves a careful study of the error propagation, cut-off of the nonlinearity and the energy method. Numerical examples are provided to confirm our theoretical analysis.