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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References
Abramowitz M, Stegun I-A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications, 1972
Bruno O, Geuzaine C, Monro J, et al. Prescribed error tolerances within fixed computational times for scattering problems of arbitrarily high frequency: The convex case. Philos Trans R Soc Lond Ser A Math Phys Eng Sci, 2004, 362: 629–645
Chandler-Wilde S-N, Graham I-G, Langdon S, et al. Numerical-asymptotic boundary integral methods in highfrequency acoustic scattering. Acta Numer, 2012, 21: 89–305
Chung K-C, Evans G-A, Webster J-R. A method to generate generalized quadrature rules for oscillatory integrals. Appl Numer Math, 2000, 34: 85–93
Colton D, Kress R. Integral Equation Methods in Scattering Theory. New York: Wiley, 1983
Deaño A, Huybrechs D, Iserles A. Computing Highly Oscillatory Integrals. Philadelphia: SIAM, 2018
Domínguez V, Graham I-G, Kim T. Filon-Clenshaw-Curtis rules for highly oscillatory integrals with algebraic singularities and stationary points. SIAM J Numer Anal, 2013, 51: 1542–1566
Erdelyi A. Asymptotic representations of Fourier integrals and the method of stationary phase. J Soc Ind Appl Math, 1955, 3: 17–27
Gao J, Condon M, Iserles A. Quadrature methods for highly oscillatory singular integrals. http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2016_02.pdf, 2016
Gao J, Iserles A. A generalization of Filon-Clenshaw-Curtis quadrature for hihgly oscillatory integrals. BIT, 2017, 4: 1–19
He G, Xiang S, Zhu E. Efficient computation of highly oscillatory integrals with weak singularities by Gauss-type method. Int J Comput Math, 2014, 93: 1–25
Huybrechs D, Olver S. Highly oscillatory quadrature. In: Highly Oscillatory Problems. Cambridge: Cambridge University Press, 2009, 25–50
Huybrechs D, Vandewalle S. On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J Numer Anal, 2006, 44: 1026–1048
Iserles A. On the numerical quadrature of highly-oscillating integrals i: Fourier transforms. IMA J Numer Anal, 2004, 24: 365–391
Iserles A, Nørsett S-P. On quadrature methods for highly oscillatory integrals and their implementation. BIT, 2004, 44: 755–772
Iserles A, Nørsett S-P. Efficient quadrature of highly oscillatory integrals using derivatives. Proc R Soc Lond Ser A Math Phy Eng Sci, 2005, 461: 1383–1399
Kang H, Xiang S. Efficient integration for a class of highly oscillatory integrals. Appl Math Comput, 2011, 218: 3553–3564
Kang H, Xiang S, He G. Computation of integrals with oscillatory and singular integrands using Chebyshev expansions. J Comput Appl Math, 2013, 242: 141–156
Levin D. Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations. Math Comp, 1982, 38: 531–538
Li J, Wang X, Wang T. A universal solution to one-dimensional oscillatory integrals. Sci China Ser F, 2008 51: 1614–1622
Lyness J, Lottes J. Asymptotic expansions for oscillatory integrals using inverse functions. BIT, 2009, 49: 397–417
Ma Y, Xu Y. Computing highly oscillatory integrals. Math Comp, 2017, 87: 309–345
Olver S. Moment-free numerical integration of highly oscillatory functions. IMA J Numer Anal, 2006, 26: 213–227
Olver S. Fast, numerically stable computation of oscillatory integrals with stationary points. BIT, 2010, 50: 149–171
Olver S. Shifted GMRES for oscillatory integrals. Numer Math, 2010, 114: 607–628
Piessens R, Branders M. On the computation of Fourier transforms of singular functions. J Comput Appl Math, 1992, 43: 159–169
Shen J, Tang T, Wang L. Spectral Methods: Algorithms, Analysis and Applications. Heidelberg: Springer, 2011
Spence E-A, Kamotski I-V, Smyshlyaev V-P. Coercivity of combined boundary integral equations in high-frequency scattering. Comm Pure Appl Math, 2014, 68: 1587–1639
Wang Y, Xiang S. tA Levin method for logarithmically singular oscillatory integrals. ArXiv:1901.05192, 2019
Xiang S. Efficient Filon-type methods for ∫abf(x) eiwg(x) dx. Numer Math, 2007, 105: 633–658
Xiang S. Numerical analysis of a fast integration method for highly oscillatory functions. BIT, 2007, 47: 469–482
Xiang S. On the Filon and Levin methods for highly oscillatory integral. J Comput Appl Math, 2007, 208: 434–439
Xiang S, Chen X, Wang H. Error bounds for approximation in Chebyshev points. Numer Math, 2010, 116: 463–491
Xiang, He G, Cho Y. On error bounds of Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals. Adv Comput Math, 2014, 41: 573–597
Xu Z, Milovanović G-V, Xiang S. Efficient computation of highly oscillatory integrals with Hankel kernel. Appl Math Comput, 2015, 261: 312–322
Xu Z, Xiang S. Gauss-type quadrature for highly oscillatory integrals with algebraic singularities and applications. Int J Comput Math, 2017, 94: 1123–1137
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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Wang, Y., Xiang, S. Levin methods for highly oscillatory integrals with singularities. Sci. China Math. 65, 603–622 (2022). https://doi.org/10.1007/s11425-018-1626-x
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DOI: https://doi.org/10.1007/s11425-018-1626-x