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An Adaptive Multiresolution Ultra-weak Discontinuous Galerkin Method for Nonlinear Schrödinger Equations

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Abstract

This paper develops a high-order adaptive scheme for solving nonlinear Schrödinger equations. The solutions to such equations often exhibit solitary wave and local structures, which make adaptivity essential in improving the simulation efficiency. Our scheme uses the ultra-weak discontinuous Galerkin (DG) formulation and belongs to the framework of adaptive multiresolution schemes. Various numerical experiments are presented to demonstrate the excellent capability of capturing the soliton waves and the blow-up phenomenon.

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Acknowledgements

We would like to thank Qi Tang and Kai Huang for the assistance and discussion in code implementation.

Funding

Y. Liu: Research supported in part by a grant from the Simons Foundation (426993, Yuan Liu). W. Guo: Research is supported by NSF grant DMS-1830838. Y. Cheng: Research is supported by NSF grants DMS-1453661 and DMS-1720023. Z. Tao: Research is supported by NSFC Grant 12001231.

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Correspondence to Juntao Huang.

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Tao, Z., Huang, J., Liu, Y. et al. An Adaptive Multiresolution Ultra-weak Discontinuous Galerkin Method for Nonlinear Schrödinger Equations. Commun. Appl. Math. Comput. 4, 60–83 (2022). https://doi.org/10.1007/s42967-020-00096-0

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  • DOI: https://doi.org/10.1007/s42967-020-00096-0

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