Abstract
The modified Filon–Clenshaw–Curtis rules, proposed earlier by the author, are combined with the (double) exponential transformations in such a way that (1) the necessity of computing the inverse of the oscillator function is released, (2) possible inaccuracy due to rounding error, when the amplitude function has endpoint singularities, is treated, (3) difficulties raised by the stationary points of the oscillator function are treated, and (4) all the benefits of the original Filon–Clenshaw–Curtis rules are preserved. We also carry out some numerical experiments, which illustrate the efficiency of the proposed algorithms while earlier methods based on Filon–Clenshaw–Curtis rules result in poor approximations or even fail.
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Intel Core i7-8550 CPU with a clock speed between 1.80 GHz and 1.99 GHz with 8 GB of RAM.
References
Asheim, A., Huybrechs, D.: Extraction of uniformly accurate phase functions across smooth shadow boundaries in high frequency scattering problems. SIAM J. Appl. Math. 74, 454–476 (2014)
Berrut, J.P., Trefethen, L.: Barycentric Lagrange interpolation. SIAM Rev. 46, 501–517 (2004)
Chandler-Wilde, S., Graham, I., Langdon, S., Spence, E.: Numerical-asymptotic boundary integral methods in high-frequency scattering. Acta Numerica 21, 89–305 (2012)
Chandler-Wilde, S., Langdon, S.: A Galerkin boundary element method for high frequency scattering by convex polygons. SIAM J. Numer. Anal. 45, 610–640 (2007)
Deaño, A., Huybrechs, D.: Complex Gaussian quadrature of oscillatory integrals. Numer. Math. 112, 197–219 (2009)
Domínguez, V., Graham, I., Kim, T.: Filon–Clenshaw–Curtis rules for highly oscillatory integrals with algebraic singularities and stationary points. SIAM J. Numer. Anal. 51, 1542–1566 (2013)
Domínguez, V., Graham, I., Smyshlyaev, V.: Stability and error estimates for Filon–Clenshaw–Curtis rules for highly oscillatory integrals. IMA J. Numer. Anal. 31, 1253–1280 (2011)
Gao, J., Iserles, A.: A generalization of Filon–Clenshaw–Curtis quadrature for highly oscillatory integrals. BIT 57, 943–961 (2017)
Higham, N.: The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal. 24, 547–556 (2004)
Huybrechs, D., Vandewalle, S.: On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J. Numer. Anal. 44, 1026–1048 (2006)
Iserles, A., Nørsett, S.: On quadrature methods for highly oscillatory integrals and their implementation. BIT 44, 755–772 (2004)
Iserles, A., Nørsett, S.: Efficient quadrature of highly oscillatory integrals using derivatives. Proc. R. Soc. Lond. A 461, 1383–1399 (2005)
Kim, T.: Asymptotic and numerical methods for high-frequency scattering problems. Ph.D. thesis, University of Bath (2012)
Levin, D.: Fast integration of rapidly oscillatory functions. J. Comput. Appl. Math. 67, 95–101 (1996)
Majidian, H.: Modified Filon–Clenshaw–Curtis rules for oscillatory integrals with a nonlinear oscillator. Preprint available at arXiv:1604.05074
Majidian, H.: Stable application of Filon–Clenshaw–Curtis rules to singular oscillatory integrals by exponential transformations. BIT 59, 155–181 (2019)
Mori, M.: An IMT-type double exponential formula for numerical integration. Publ. Res. Inst. Math. Sci. 14, 713–729 (1978)
Mori, M.: Discovery of the double exponential transformation and its developments. Publ. Res. Inst. Math. Sci. 41, 897–935 (2005)
Mori, M., Sugihara, M.: The double-exponential transformation in numerical analysis. J. Comput. Appl. Math. 127, 287–296 (2001)
Rutishauser, H.: Vorlesungen über numerische Mathematik, vol. 1. Birkhäuser, Basel, Stuttgart, 1976; English translation, Lectures on Numerical Mathematics, Walter Gautschi, ed. Birkhäuser, Boston (1990)
Spence, E., Chandler-Wilde, S., Graham, I., Smyshlyaev, V.: A new frequency-uniform coercive boundary integral equation for acoustic scattering. Commun. Pure Appl. Math. 64, 1384–1415 (2011)
Takahasi, H., Mori, M.: Quadrature formulas obtained by variable transformation. Numer. Math. 21, 206–219 (1973)
Takahasi, H., Mori, M.: Double exponential formulas for numerical integration. Publ. Res. Inst. Math. Sci. 9, 721–741 (1974)
Xiang, S.: Efficient Filon-type methods for \(\int ^b_af(x)e^{{\rm i}\omega g(x)}\,{\rm d} x\). Numer. Math. 105, 633–658 (2007)
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Communicated by Christian Lubich.
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Majidian, H. Efficient computation of oscillatory integrals by exponential transformations. Bit Numer Math 61, 1337–1365 (2021). https://doi.org/10.1007/s10543-021-00855-2
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DOI: https://doi.org/10.1007/s10543-021-00855-2