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In this paper, we propose a family of time-stepping schemes for approximatinggeneral nonlinear Schr\"odinger equations. The proposed schemes all satisfyboth mass conservation and energy conservation. Truncation and dispersion erroranalyses are provided for each proposed scheme. Efficient fixed-point iterativesolvers are also constructed to solve the resulting nonlinear discreteproblems. As a byproduct, an efficient one-step implementation of the BDFschemes is obtained as well. Extensive numerical experiments are presented todemonstrate the convergence and the capability of capturing the blow-upphenomenon of the proposed schemes.

Mass- and energy-conserved numerical schemes for nonlinear Schr\"odinger equations