Conerservative and Nonconservative Schemes for the Solution of the Nonlinear Schrödinger Equation

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Abstract
Five methods for the integration in time of a semidiscretization of the nonlinear Schrödinger equation are extensively tested. Three of them (a partly explicit scheme and two splitting procedures) are found to perform poorly. The reasons for their failure, including the so-called nonlinear blow-up, are analysed. We draw general conclusions on the advantages and drawbacks associated with the use of time-integrators which exactly conserve energy.