On the effectiveness of spectral methods for the numerical solution of multi-frequency highly oscillatory Hamiltonian problems

On the effectiveness of spectral methods for the numerical solution of multi-frequency highly... Numer Algor https://doi.org/10.1007/s11075-018-0552-9 ORIGINAL PAPER On the effectiveness of spectral methods for the numerical solution of multi-frequency highly oscillatory Hamiltonian problems 1 2 Luigi Brugnano · Juan I. Montijano · Luis Randez ´ Received: 23 November 2017 / Accepted: 24 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract Multi-frequency, highly oscillatory Hamiltonian problems derive from the mathematical modelling of many real-life applications. We here propose a variant of Hamiltonian Boundary Value Methods (HBVMs), which is able to efficiently deal with the numerical solution of such problems. We present algorithms to select the parameters of the methods that allow one to obtain numerical approximations with spectral accuracy. We also propose an efficient implementation of the methods when using a Newton-type iteration to solve the implicit equations associated with this class of formulas. Keywords Multi-frequency highly oscillatory problems · Hamiltonian problems · Energy-conserving methods · Spectral methods · Legendre polynomials · Hamiltonian Boundary Value Methods · HBVMs Mathematics Subject Classification (2010) 65P10 · 65L05 Juan I. Montijano monti@unizar.es Luigi Brugnano luigi.brugnano@unifi.it Luis Randez ´ randez@unizar.es Dipartimento di Matematica e Informatica “U. Dini”, Universita ` di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy IUMA, Universidad de Zaragoza, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Numerical Algorithms Springer Journals

On the effectiveness of spectral methods for the numerical solution of multi-frequency highly oscillatory Hamiltonian problems

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Computer Science; Numeric Computing; Algorithms; Algebra; Theory of Computation; Numerical Analysis
ISSN
1017-1398
eISSN
1572-9265
D.O.I.
10.1007/s11075-018-0552-9
Publisher site
See Article on Publisher Site

Abstract

Numer Algor https://doi.org/10.1007/s11075-018-0552-9 ORIGINAL PAPER On the effectiveness of spectral methods for the numerical solution of multi-frequency highly oscillatory Hamiltonian problems 1 2 Luigi Brugnano · Juan I. Montijano · Luis Randez ´ Received: 23 November 2017 / Accepted: 24 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract Multi-frequency, highly oscillatory Hamiltonian problems derive from the mathematical modelling of many real-life applications. We here propose a variant of Hamiltonian Boundary Value Methods (HBVMs), which is able to efficiently deal with the numerical solution of such problems. We present algorithms to select the parameters of the methods that allow one to obtain numerical approximations with spectral accuracy. We also propose an efficient implementation of the methods when using a Newton-type iteration to solve the implicit equations associated with this class of formulas. Keywords Multi-frequency highly oscillatory problems · Hamiltonian problems · Energy-conserving methods · Spectral methods · Legendre polynomials · Hamiltonian Boundary Value Methods · HBVMs Mathematics Subject Classification (2010) 65P10 · 65L05 Juan I. Montijano monti@unizar.es Luigi Brugnano luigi.brugnano@unifi.it Luis Randez ´ randez@unizar.es Dipartimento di Matematica e Informatica “U. Dini”, Universita ` di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy IUMA, Universidad de Zaragoza,

Journal

Numerical AlgorithmsSpringer Journals

Published: Jun 2, 2018

References

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