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(Spectral) Chebyshev collocation methods for solving differential equations

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Abstract

Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along the Legendre orthonormal basis. Interestingly, this approach can be extended to cope with other orthonormal bases and, in particular, we here consider the case of the Chebyshev polynomial basis. The corresponding Runge-Kutta methods were previously obtained by Costabile and Napoli (Numer. Algo., 27, 119–130 2021). In this paper, the use of a different framework allows us to carry out a novel analysis of the methods also when they are used as spectral formulae in time, along with some generalizations of the methods.

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Notes

  1. In fact, for isolated mechanical systems, H has the physical meaning of the total energy.

  2. This amounts to study HBVMs as continuous-stage Runge-Kutta methods, as it has been done in [1, 2].

  3. Hereafter, |⋅| will devote any convenient vector norm.

  4. For sake of simplicity, hereafter we shall assume f be an analytic function.

  5. Clearly, (15) is derived by considering the special case ω(c) ≡ 1.

  6. As is usual, hereafter, Tj(x), j ≥ 0, denote the Chebyshev polynomials of the first kind.

  7. Here, we have used the name of the MatlabⒸ functions implementing the two transformations.

  8. As is usual, 1 flop denotes a basic fl oating-point op eration.

  9. Clearly, when k = s, (41) reduces to \(\mathcal { I}_{s}=\mathcal { P}_{s}X_{s}\), according to Lemma 1.

  10. The second property has already been used in the proof of Lemma 3.

  11. Also this property has already been used, in the proof of Theorem 7.

  12. For sake of completeness, we also report that the former method requires a total of 106449 fixed-point iterations, whereas the latter 100642.

  13. Consequently, the considered timesteps are approximately in the interval [0.4, 1].

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Acknowledgements

The authors are indebted to the reviewers, for the careful reading of the manuscript and the precious comments and remarks.

Funding

The authors wish to thank the mrSIR project (https://www.mrsir.it/en/about-us/) for the financial support.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bari, Bari, Italy

    Pierluigi Amodio & Felice Iavernaro

  2. Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Florence, Italy

    Luigi Brugnano

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  1. Pierluigi Amodio
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Correspondence to Luigi Brugnano.

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Amodio, P., Brugnano, L. & Iavernaro, F. (Spectral) Chebyshev collocation methods for solving differential equations. Numer Algor 93, 1613–1638 (2023). https://doi.org/10.1007/s11075-022-01482-w

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