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 (Journal of Computational Mathematics, 2024)
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) discretizationhas been developed (Guo et al., 2021) and proved to be uniformly accurate about $ε$ up tothe time at $\mathcal{O}(1/ε^2).$ However, the EWIFP method is not time symmetric and can notpreserve the discrete energy. As we know, the time symmetry and energy-preservationare the important structural features of the true solution and we hope that this structurecan be inherited along the numerical solution. In this work, we propose a time symmetricand 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 $\mathcal{O}(h^{m_0} + ε^{ 2−β}τ^2)$ up to thetime at $\mathcal{O}(1/ε^β)$ for $β ∈ [0, 2],$ where $h$ and $τ$ are the mesh size and time step, respectively,and $m_0$ depends on the regularity conditions. The tools for error analysis mainly includecut-off technique and the standard energy method. We also extend the results on errorbounds, energy-preservation and time symmetry to the oscillatory NLSW with wavelengthat $\mathcal{O}(ε^2)$ in time which is equivalent to the NLSW with weak nonlinearity. Numericalexperiments confirm that the theoretical results in this paper are correct. Our method isnovel because that to the best of our knowledge there has not been any energy-preservingexponential wave integrator method for the NLSW.