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Analysis of spectral Hamiltonian boundary value methods (SHBVMs) for the numerical solution of ODE problems

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Abstract

Recently, the numerical solution of stiffly/highly oscillatory Hamiltonian problems has been attacked by using Hamiltonian boundary value methods (HBVMs) as spectral methods in time. While a theoretical analysis of this spectral approach has been only partially addressed, there is enough numerical evidence that it turns out to be very effective even when applied to a wider range of problems. Here, we fill this gap by providing a thorough convergence analysis of the methods and confirm the theoretical results with the aid of a few numerical tests.

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Notes

  1. In fact, if problem (1) is non autonomous, t can be included in the state vector.

  2. The used arguments are mainly adapted from [21].

  3. Hereafter, for sake of clarity, we shall denote by |⋅| any convenient vector or matrix norm.

  4. The proof uses arguments similar to those of [13, Theorem 4].

  5. Hereafter, \(\doteq \) means “equal within the round-off error level of the used finite precision arithmetic”.

  6. This latter criterion was that used in [18] and [9].

  7. In such a case, the machine precision is u ≈ 10− 16.

  8. i.e., \(0<c_{1}<\dots <c_{k}<1\) are the zeros of Pk.

  9. Hereafter, the initial approximation \(\hat {\boldsymbol {\gamma }}^{0}=\bf 0\) is conveniently used.

  10. i.e., factor Σ− 1.

  11. As matter of fact, considering more stringent tolerances does not improve the accuracy of the computed numerical solution.

  12. In this case, the Gauss methods exhibit a super-convergence in the conservation of the Hamiltonian (3 times the usual order) and HBVMs do the same with the angular momentum. This is due to the fact that the error is measured only at the end of each period.

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Acknowledgments

The authors are very grateful to two unknown referees, for the careful reading of the manuscript, and for their precious comments, which allowed to formulate in a cleaner and more precise way the results presented in the paper.

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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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  2. Luigi Brugnano
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  3. Felice Iavernaro
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Correspondence to Luigi Brugnano.

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Amodio, P., Brugnano, L. & Iavernaro, F. Analysis of spectral Hamiltonian boundary value methods (SHBVMs) for the numerical solution of ODE problems. Numer Algor 83, 1489–1508 (2020). https://doi.org/10.1007/s11075-019-00733-7

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