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Unconditional convergence of linearized implicit finite difference method for the 2D/3D Gross-Pitaevskii equation with angular momentum rotation

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Abstract

This paper is concerned with the time-step condition of linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. Unlike the existing studies in the literature, where the cut-off function technique was used to establish the error estimates under some conditions of the time-step size, this paper introduces an induction argument and a ‘lifting’ technique as well as some useful inequalities to build the optimal maximum error estimate without any constraints on the time-step size. The analysis method can be directly extended to the general nonlinear Schrödinger-type equations in twoand three-dimensions and other linear implicit finite difference schemes. As a by-product, this paper defines a new type of energy functional of the grid functions by using a recursive relation to prove that the proposed scheme preserves well the total mass and energy in the discrete sense. Several numerical results are reported to verify the error estimates and conservation laws.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11571181 and 11731014), Natural Science Foundation of Jiangsu Province (Grant No. BK20171454) and Qing Lan Project.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Tingchun Wang

  2. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Boling Guo

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  1. Tingchun Wang
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  2. Boling Guo
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Correspondence to Tingchun Wang.

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Wang, T., Guo, B. Unconditional convergence of linearized implicit finite difference method for the 2D/3D Gross-Pitaevskii equation with angular momentum rotation. Sci. China Math. 62, 1669–1686 (2019). https://doi.org/10.1007/s11425-016-9212-1

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  • DOI: https://doi.org/10.1007/s11425-016-9212-1

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