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 ǫ 2p with ǫ ∈ (0, 1] a dimensionless parameter and p ∈ + . When 0 < ǫ ≪ 1, the long-time dynamics of the problem is equivalent to that of the NLSW with(1)-nonlinearity and (ǫ)-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 (h m−1 + ǫ 2p−β τ 2 ) up to the time at (1/ǫ β ) with 0 ≤ β ≤ 2p, the mesh size h, time step τ 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.