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Two energy-conserving and compact finite difference schemes for two-dimensional Schrödinger-Boussinesq equations

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Abstract

In this paper, we study two compact finite difference schemes for the Schrödinger-Boussinesq (SBq) equations in two dimensions. The proposed schemes are proved to preserve the total mass and energy in the discrete sense. In our numerical analysis, besides the standard energy method, a “cut-off” function technique and a “lifting” technique are introduced to establish the optimal H1 error estimates without any restriction on the grid ratios. The convergence rate is proved to be of O(τ2 + h4) with the time step τ and mesh size h. In addition, a fast finite difference solver is designed to speed up the numerical computation of the proposed schemes. The numerical results are reported to verify the error estimates and conservation laws.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions that helped greatly to improved the quality of this article.

Funding

This work is financially supported by the National Natural Science Foundation of China under Grant No.11571181.

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Correspondence to Tingchun Wang.

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Liao, F., Zhang, L. & Wang, T. Two energy-conserving and compact finite difference schemes for two-dimensional Schrödinger-Boussinesq equations. Numer Algor 85, 1335–1363 (2020). https://doi.org/10.1007/s11075-019-00867-8

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