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Levin methods for highly oscillatory integrals with singularities

Abstract

In this paper, new Levin methods are presented for calculating oscillatory integrals with algebraic and/or logarithmic singularities. To avoid singularity, the technique of singularity separation is applied and then the singular ODE occurring in classic Levin methods is converted into two kinds of non-singular ODEs. The solutions of one can be obtained explicitly, while those of the other can be solved efficiently by collocation methods. The proposed methods can attach arbitrarily high asymptotic orders and also enjoy superalgebraic convergence with respect to the number of collocation points. Several numerical experiments are presented to validate the efficiency of the proposed methods.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11771454) and Research Fund of National University of Defense Technology (Grant No. ZK19-19). The authors are grateful to the referees’ helpful suggestions and insightful comments, which helped improve the manuscript significantly. The authors thank Dr. Saira and Dr. Suliman at Central South University for their careful checking of numerous details.

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Correspondence to Shuhuang Xiang.

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Wang, Y., Xiang, S. Levin methods for highly oscillatory integrals with singularities. Sci. China Math. (2020). https://doi.org/10.1007/s11425-018-1626-x

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Keywords

  • Levin method
  • highly oscillatory integral
  • algebraic singularity
  • logarithmic singularity

MSC(2010)

  • 65D30
  • 65D32
  • 65L99