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                <forename type="first">Xavier</forename>
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            <funder>This work was partially supported by NSF DMS-2012585 and AFOSR FA9550-20-1-0309 (J. Shen), the Inria through the associated Inria team project BEC2HPC (X. Antoine and Q. Tang), the Fundamental Research Funds for the Central Universities (No. YJ201807) and the National Natural Science Foundation of China (No. 11971335) (Q. Tang).</funder>
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                <term xml:lang="en">pseudospectral method</term>
                <term xml:lang="en">Lagrange multiplier</term>
                <term xml:lang="en">Scalar Auxiliary Variables</term>
                <term xml:lang="en">explicit time scheme</term>
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              <p>In this paper, based on the Scalar Auxiliary Variable (SAV) approach and a newly proposed Lagrange multiplier (LagM) approach originally constructed for gradient flows, we propose two linear implicit pseudo-spectral schemes for simulating the dynamics of general nonlinear Schrödinger/Gross-Pitaevskii equations. Both schemes are of spectral/second-order accuracy in spatial/temporal direction. The SAV based scheme preserves a modified total energy and approximate the mass to third order (with respect to time steps), while the LagM based scheme could preserve exactly the mass and original total energy. A nonlinear algebraic system has to be solved at every time step for the LagM based scheme, hence the SAV scheme is usually more efficient than the LagM one. On the other hand, the LagM scheme may outperform the SAV ones in the sense that it conserves the original total energy and mass and usually admits smaller errors. Ample numerical results are presented to show the effectiveness, accuracy and performance of the proposed schemes.</p>
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