Optimal error estimate of a conformal Fourier pseudo-spectral method for the damped nonlinear Schrödinger equation. (English) Zbl 1407.65215
Summary: In this article, a Fourier pseudospectral method, which preserves the conforal conservation la, is proposed for solving the damped nonlinear Schrödinger equation. Based on the energy method and the semi-norm equivalence between the Fourier pseudospectral method and the finite difference method, a priori estimate for the new method is established, which shows that the proposed method is unconditionally convergent with order of \(O(\tau^2 + J^{1-r})\) in the discrete \(L^{\infty}\)-norm, where \(\tau\) is the time step and \(J\) is the number of collocation points used in the spectral method. Some numerical results are addressed to confirm our theoretical analysis.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65T50 | Numerical methods for discrete and fast Fourier transforms |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 47H10 | Fixed-point theorems |
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