Chebyshev collocation method for solving second order ODEs using integration matrices

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Abstract

The spectral collocation method for solving two-point boundary value problems for second order differential equations is implemented, based on representing the solution as an expansion in Chebyshev polynomials. The approach allows a stable calculation of both the spectral representation of the solution and its pointwise representation on any required grid in the definition domain of the equation and additional conditions of the multipoint problem. For the effective construction of SLAE, the solution of which gives the desired coefficients, the Chebyshev matrices of spectral integration are actively used. The proposed algorithms have a high accuracy for moderate-dimension systems of linear algebraic equations. The matrix of the system remains well-conditioned and, with an increase in the number of collocation points, allows finding solutions with ever-increasing accuracy.

About the authors

Konstantin P. Lovetskiy

RUDN University

Email: lovetskiy-kp@rudn.ru
ORCID iD: 0000-0002-3645-1060

Candidate of Sciences in Physics and Mathematics, Associate Professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Dmitry S. Kulyabov

RUDN University; Joint Institute for Nuclear Research

Author for correspondence.
Email: kulyabov-ds@rudn.ru
ORCID iD: 0000-0002-0877-7063

Professor, Doctor of Sciences in Physics and Mathematics, Professor at the Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University); Senior Researcher of Laboratory of Information Technologies, Joint Institute for Nuclear Research

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation

Leonid A. Sevastianov

RUDN University; Joint Institute for Nuclear Research

Email: sevastianov-la@rudn.ru
ORCID iD: 0000-0002-1856-4643

Professor, Doctor of Sciences in Physics and Mathematics, Professor at the Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University); Leading Researcher of Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation

Stepan V. Sergeev

RUDN University

Email: 1142220124@rudn.ru
ORCID iD: 0009-0004-1159-4745

PhD student of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

References

  1. V. A. Soifer, Diffraction computer optics [Difraktsionnaya komp’yuternaya optika]. M.: FIZMATLIT, 2007, in Russian.
  2. A. A. Egorov and L. A. Sevastianov, “Structure of modes of a smoothly irregular integrated-optical four-layer three-dimensional waveguide,” Quantum Electronics, vol. 39, no. 6, pp. 566-574, Jun. 2009. DOI: 10. 1070/QE2009v039n06ABEH013966.
  3. A. L. Sevastianov, “Asymptotic method for constructing a model of adiabatic guided modes of smoothly irregular integrated optical waveguides,” Discrete and Continuous Models and Applied Computational Science, vol. 28, no. 3, pp. 252-273, 2020. doi: 10.22363/2658-4670-2020-28-3-252-273.
  4. A. L. Sevastianov, “Single-mode propagation of adiabatic guided modes in smoothly irregular integral optical waveguides,” Discrete and Continuous Models and Applied Computational Science, vol. 28, no. 4, pp. 361-377, 2020. doi: 10.22363/2658-4670-2020-28-4-361-377.
  5. L. Greengard, “Spectral integration and two-point boundary value problems,” SIAM Journal on Numerical Analysis, vol. 28, no. 4, pp. 1071-1080, 1991. doi: 10.1137/0728057.
  6. A. Amiraslani, R. M. Corless, and M. Gunasingam, “Differentiation matrices for univariate polynomials,” Numerical Algorithms, vol. 83, no. 1, pp. 1-31, 2020. doi: 10.1007/s11075-019-00668-z.
  7. J. P. Boyd, Chebyshev and Fourier spectral methods, second. Dover Books on Mathematics, 2013.
  8. J. C. Mason and D. C. Handscomb, Chebyshev polynomials. New York: Chapman and Hall/CRC Press, 2002. doi: 10.1201/9781420036114.
  9. S. E. El-gendi, “Chebyshev solution of differential, integral and integrodifferential equations,” The Computer Journal, vol. 12, no. 3, pp. 282-287, Aug. 1969. doi: 10.1093/comjnl/12.3.282.
  10. L. N. Trefethen, “Is Gauss quadrature better than Clenshaw-Curtis?” SIAM Review, vol. 50, no. 1, pp. 67-87, 2008. doi: 10.1137/060659831.
  11. L. A. Sevastianov, K. P. Lovetskiy, and D. S. Kulyabov, “A new approach to the formation of systems of linear algebraic equations for solving ordinary differential equations by the collocation method [Novyy podkhod k formirovaniyu sistem lineynykh algebraicheskikh uravneniy dlya resheniya obyknovennykh differentsial’nykh uravneniy metodom kollokatsiy],” Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, vol. 23, no. 1, pp. 36-47, 2023, in Russian. doi: 10.18500/1816-9791-2023-23-1-36-47.
  12. N. Egidi and P. Maponi, “A spectral method for the solution of boundary value problems,” Applied Mathematics and Computation, vol. 409, p. 125 812, 2021. doi: 10.1016/j.amc.2020.125812.
  13. H. B. Keller, Numerical methods for two-point boundary value problems. Boston: Ginn-Blaisdell, 1968.
  14. D. Gottlieb and S. A. Orszag, Numerical analysis of spectral methods. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1977.
  15. J. F. Epperson, An introduction to numerical methods and analysis, second. John Wiley & Sons, Inc, 2013.
  16. X. Zhang and J. P. Boyd, Asymptotic coefficients and errors for Chebyshev polynomial approximations with weak endpoint singularities: effects of different bases, 2022. doi: 10.48550/arXiv.2103.11841.
  17. J. P. Boyd and D. H. Gally, “Numerical experiments on the accuracy of the Chebyshev-Frobenius companion matrix method for finding the zeros of a truncated series of Chebyshev polynomials,” Journal of Computational and Applied Mathematics, vol. 205, no. 1, pp. 281-295, 2007. doi: 10.1016/j.cam.2006.05.006.
  18. B. Fornberg, A practical guide to pseudospectral methods. New York: Cambridge University Press, 1996.
  19. F. Rezaei, M. Hadizadeh, R. Corless, and A. Amiraslani, “Structural analysis of matrix integration operators in polynomial bases,” Banach Journal of Mathematical Analysis, vol. 16, no. 1, p. 5, 2022. DOI: 10. 1007/s43037-021-00156-4.
  20. L. C. Young, “Orthogonal collocation revisited,” Computer methods in Applied Mechanics and Engineering, vol. 345, pp. 1033-1076, 2019. doi: 10.1016/j.cma.2018.10.019.
  21. S. Olver and A. Townsend, “A fast and well-conditioned spectral method,” SIAM Review, vol. 55, no. 3, pp. 462-489, 2013. doi: 10.1137/120865458.
  22. M. Planitz et al., Numerical recipes: the art of scientific computing, 3rd Edition. New York: Cambridge University Press, 2007.
  23. L. A. Sevastianov, K. P. Lovetskiy, and D. S. Kulyabov, “Multistage collocation pseudo-spectral method for the solution of the first order linear ODE,” in 2022 VIII International Conference on Information Technology and Nanotechnology (ITNT), 2022, pp. 1-6. doi: 10.1109/ITNT55410.2022.9848731.

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