Efficient dissipation-preserving approaches for the damped nonlinear Schrödinger equation

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In this paper, we construct two dissipation-preserving prediction-correction schemes for the damped nonlinear Schrödinger equation. The temporal second-order scheme is implicit, but it only requires solving some scalar nonlinear equations in the prediction step plus a scalar quadratic equation in the correction step. The temporal fourth-order scheme is fully explicit. The compact representation of the schemes together with the discrete fast Fourier transform reduces the storage requirement and CPU cost significantly. Ample numerical results are reported to validate the effectiveness, robustness and accuracy of the proposed schemes. Numerical results also show that the explicit fourth-order scheme is more efficient and accurate than the existing schemes.