Uniform error bound of an exponential wave integrator for the long-time dynamics of the nonlinear Schrödinger equation with wave operator
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We establish the 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 ε^2p with ε∈ (0, 1] a dimensionless parameter and p ∈ℕ^+. When 0 < ε≪ 1, the long-time dynamics of the problem is equivalent to that of the NLSW with O(1)-nonlinearity and O(ε)-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 O(h^m-1+ε^2p-βτ^2) up to the time at O(1/ε^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.
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