TY - JOUR
T1 - Uniform Error Bounds of an Energy-Preserving Exponential Wave Integrator Fourier Pseudo-Spectral Method for the Nonlinear Schrödinger Equation with Wave Operator and Weak Nonlinearity
AU - Li , Jiyong
JO - Journal of Computational Mathematics
VL - 2
SP - 280
EP - 314
PY - 2024
DA - 2024/11
SN - 43
DO - http://doi.org/10.4208/jcm.2310-m2022-0141
UR - https://global-sci.org/intro/article_detail/jcm/23539.html
KW - Nonlinear Schrödinger equation with wave operator and weak nonlinearity, Fourier pseudo-spectral method, Exponential wave integrator, Energy-preserving method, Error estimates, Oscillatory problem.
AB - <p style="text-align: justify;">Recently, the numerical methods for long-time dynamics of PDEs with weak nonlinearity have received more and more attention. For the nonlinear Schrödinger equation
(NLS) with wave operator (NLSW) and weak nonlinearity controlled by a small value $ε ∈ (0, 1],$ an exponential wave integrator Fourier pseudo-spectral (EWIFP) discretization
has been developed (Guo et al., 2021) and proved to be uniformly accurate about $ε$ up to
the time at $\mathcal{O}(1/ε^2
).$ However, the EWIFP method is not time symmetric and can not
preserve the discrete energy. As we know, the time symmetry and energy-preservation
are the important structural features of the true solution and we hope that this structure
can be inherited along the numerical solution. In this work, we propose a time symmetric
and energy-preserving exponential wave integrator Fourier pseudo-spectral (SEPEWIFP)
method for the NLSW with periodic boundary conditions. Through rigorous error analysis,
we establish uniform error bounds of the numerical solution at $\mathcal{O}(h^{m_0} + ε^{ 2−β}\tau^2)$ up to the
time at $\mathcal{O}(1/ε^β)$ for $β ∈ [0, 2],$ where $h$ and $\tau$ are the mesh size and time step, respectively,
and $m_0$ depends on the regularity conditions. The tools for error analysis mainly include
cut-off technique and the standard energy method. We also extend the results on error
bounds, energy-preservation and time symmetry to the oscillatory NLSW with wavelength
at $\mathcal{O}(ε^2)$ in time which is equivalent to the NLSW with weak nonlinearity. Numerical
experiments confirm that the theoretical results in this paper are correct. Our method is
novel because that to the best of our knowledge there has not been any energy-preserving
exponential wave integrator method for the NLSW.</p>