Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

An Enhanced Chebyshev Collocation Method Based on the Integration of Chebyshev Interpolation

  • Kim, Philsu (Department of Mathematics, Kyungpook National University)
  • Received : 2016.03.11
  • Accepted : 2016.07.05
  • Published : 2017.06.23
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we develop an enhanced Chebyshev collocation method based on an integration scheme of the generalized Chebyshev interpolations for solving stiff initial value problems. Unlike the former error embedded Chebyshev collocation method (CCM), the enhanced scheme calculates the solution and its truncation error based on the interpolation of the derivative of the true solution and its integration. In terms of concrete convergence and stability analysis, the constructed algorithm turns out to have the $7^{th}$ convergence order and the A-stability without any loss of advantages for CCM. Throughout a numerical result, we assess the proposed method is numerically more efficient compared to existing methods.

Keywords

References

  1. https://www.dm.uniba.it/-testset/testsetivpsolvers
  2. K. Astala, L. Paivarinta, J. M. Reyes and S. Siltanen, Nonlinear Fourier analysis for discontinuous conductivities: Computational results, J. Comput. Phys., 276(2014), 74-91. https://doi.org/10.1016/j.jcp.2014.07.032
  3. A. Bourlioux, A. T. Layton and M. L. Minion, High-Order Multi-Implicit Spectral Deferred Correction Methods for Problems of Reactive Flow, J. Comput. Phys., 189(2003), 651-675. https://doi.org/10.1016/S0021-9991(03)00251-1
  4. L. Brugnano and C. Magherini, Blended implementation of block implicit methods for ODEs, Appl. Numer. Math., 42(2002), 29-45. https://doi.org/10.1016/S0168-9274(01)00140-4
  5. M. P. Calvo and E. Hairer, Accurate long-term integration of dynamical systems, Appl. Num. Math., 18(1995), 95-105. https://doi.org/10.1016/0168-9274(95)00046-W
  6. C. W. Clenshaw, The numerical solution of ordinary differential equations in Chebyshev series, in: P.I.C.C. symposium on Differential and Integral Equations, Birkhauser Verlag, Rome, 1960, p. 222.
  7. C.W. Clenshaw and H.J. Norton, The solution of nonlinear ordinary differential equations in Chebyshev series, Comput. J., 6(1963), 88-92. https://doi.org/10.1093/comjnl/6.1.88
  8. C. W. Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, 1971.
  9. E. Hairer and G. Wanner, Solving ordinary differential equations. II Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics, Springer, 1996.
  10. T. Hasegawa, T. Torii and I. Ninomiya, Generalized Chebyshev interpolation and its application to automatic quadrature, Math. Comp., 41(1983), 537-553. https://doi.org/10.1090/S0025-5718-1983-0717701-6
  11. T. Hasegawa, T. Torii and H. Sugiura, An algorithm based on the FFT for a generalized Chebyshev interpolation, Math. Comp., 54(189)(1990), 195-210. https://doi.org/10.1090/S0025-5718-1990-0990599-0
  12. P. Kim, A Chebyshev quadrature rule for one sided finite part integrals, J. Approx. Theory, 111(2)(2001), 196-219. https://doi.org/10.1006/jath.2001.3572
  13. P. Kim and U. J. Choi, Two trigonometric quadrature formulae for evaluating hypersingular integrals, Int. J. Numer. Method Eng., 56(3)(2003), 469-486. https://doi.org/10.1002/nme.582
  14. P. Kim, X. Piao and S. D. Kim, An error corrected Euler method for solving stiff problems based on Chebyshev collocation, SIAM J. Numer. Anal., 49(6)(2011), 2211-2230. https://doi.org/10.1137/100808691
  15. P. Kim, J. Kim, W. Jung and S. Bu, An error embedded method based on generalized Chebysehv polynomials, J. Comp. Phys., 306(2016), 55-72. https://doi.org/10.1016/j.jcp.2015.11.021
  16. M. A. Kopera and F. X. Giraldo, Analysis of adaptive mesh refinement for IMEX discontinuous Galerkin solutions of the compressible Euler equations with application to atmospheric simulations, J. Comput. Physics, 275 (2014), 92-117. https://doi.org/10.1016/j.jcp.2014.06.026
  17. C. Lanczos, Applied Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1956.
  18. A. T. Layton and M. L. Minion, Conservative Multi-Implicit Spectral Deferred Correction Methods for Reacting Gas Dynamics, J. Comput. Physics, 194(2)(2004), 679-715.
  19. H. Moon, F. L. Teixeira and B. Donderici, Stable pseudoanalytical computation of electromagnetic fields from arbitrarily-oriented dipoles in cylindrically stratified media, J. Comput. Physics., 273(2014), 118-142. https://doi.org/10.1016/j.jcp.2014.05.006
  20. H. Ramos and J. Vigo-Aguiar, A fourth order Runge-Kutta method based on BDF-type Chebyshev approximations, J. Comp. Appl. Math., 204(1)(2007), 124-136. https://doi.org/10.1016/j.cam.2006.04.033
  21. K.Wright, Chebyshev collocation methods for ordinary differential equations, Comput. J., 6(1963), 358-363.