Uniform Error Bound of an Exponential Wave Integrator for Long-Time Dynamics of the Nonlinear Schrödinger Equation with Wave Operator

Abstract

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 $\varepsilon\in(0,1]$ 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 $\mathscr{O}(1)$-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 $\mathscr{O}(h^{m-1}+\varepsilon^{2p-\beta}\tau^2)$ up to the time at $\mathscr{O}(1/\varepsilon^\beta)$ with $0\le \beta \le 2p$, 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.