Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Buchtitelbild

2018 | OriginalPaper | Buchkapitel

1. Functionally Fitted Continuous Finite Element Methods for Oscillatory Hamiltonian Systems

verfasst von : Xinyuan Wu, Bin Wang

Erschienen in: Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations

Verlag: Springer Singapore

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In recent decades, the numerical simulation for nonlinear oscillators has received much attention and a large number of integrators for oscillatory problems have been developed. In this chapter, based on the continuous finite element approach, we propose and analyse new energy-preserving functionally-fitted, in particular, trigonometrically-fitted methods of an arbitrarily high order for solving oscillatory nonlinear Hamiltonian systems with a fixed frequency. In order to implement these new methods in an accessable and efficient style, they are formulated as a class of continuous-stage Runge–Kutta methods. The numerical results demonstrate the remarkable accuracy and efficiency of the new methods compared with the existing high-order energy-preserving methods in the literature.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Nächstes Kapitel Exponential Average-Vector-Field Integrator for Conservative or Dissipative Systems
Literatur
1.
Betsch, P., Steinmann, P.: Inherently energy conserving time finite element methods for classical mechanics. J. Comput. Phys. 160, 88–116 (2000)MathSciNetCrossRef
2.
Bettis, D.G.: Numerical integration of products of Fourier and ordinary polynomials. Numer. Math. 14, 424–434 (1970)MathSciNetCrossRef
3.
Bottasso, C.L.: A new look at finite elements in time : a variational interpretation of Runge-Kutta methods. Appl. Numer. Math. 25, 355–368 (1997)MathSciNetCrossRef
4.
Brugnano, L., Iavernaro, F., Trigiante, D.: Hamiltonan boundary value methods (Energy preserving discrete line integral methods). J. Numer. Anal. Ind. Appl. Math. 5, 13–17 (2010)MATH
5.
Brugnano, L., Iavernaro, F., Trigiante, D.: A simple framework for the derivation and analysis of effective one-step methods for ODEs. Appl. Math. Comput. 218, 8475–8485 (2012)MathSciNetMATH
6.
Brugnano, L., Iavernaro, F., Trigiante, D.: Energy- and quadratic invariants-preserving integrators based upon Gauss-collocation formulae. SIAM J. Numer. Anal. 50, 2897–2916 (2012)MathSciNetCrossRef
7.
Calvo, M., Franco, J.M., Montijano, J.I., Rández, L.: Structure preservation of exponentially fitted Runge-Kutta methods. J. Comput. Appl. Math. 218, 421–434 (2008)MathSciNetCrossRef
8.
Calvo, M., Franco, J.M., Montijano, J.I., Rández, L.: Symmetric and symplectic exponentially fitted Runge-Kutta methods of high order. Comput. Phys. Commun. 181, 2044–2056 (2010)MathSciNetCrossRef
9.
Calvo, M., Franco, J.M., Montijano, J.I., Rández, L.: On high order symmetric and symplectic trigonometrically fitted Runge-Kutta methods with an even number of stages. BIT Numer. Math. 50, 3–21 (2010)MathSciNetCrossRef
10.
Celledoni, E., Mclachlan, R.I., Mclaren, D.I., Owren, B., Quispel, G.R.W., Wright, W.M.: Energy-preserving Runge-Kutta methods. ESIAM. Math. Model. Numer. Anal. 43, 645–649 (2009)MathSciNetCrossRef
11.
Celledoni, E., Mclachlan, R.I., Owren, B., Quispel, G.R.W.: Energy-preserving integrators and the structure of B-series. Found. Comput. Math. 10, 673–693 (2010)MathSciNetCrossRef
12.
Celledoni, E., Grimm, V., Mclachlan, R.I., Mclaren, D.I., O’Neale, D., Owren, B., Quispel, G.R.W.: Preserving energy resp. dissipation in numerical PDEs using the ‘Average Vector Field’ method. J. Comput. Phys. 231, 6770–6789 (2012)MathSciNetCrossRef
13.
Chen, J.B., Qin, M.Z.: Multisymplectic fourier pseudospectral method for the nonlinear Schrödinger equation. Electron. Trans. Numer. Anal. 12, 193–204 (2001)MathSciNetMATH
14.
Cohen, D., Jahnke, T., Lorenz, K., Lubich, C.: Numerical integrators for highly oscillatory Hamiltonian systems: a review. In: Mielke, A. (ed.) Analysis, Modeling and Simulation of Multiscale Problems, pp. 553–576. Springer, Berlin (2006)CrossRef
15.
Coleman, J.P.: P-stability and exponential-fitting methods for \(y^{\prime \prime }=f(x, y)\). IMA J. Numer. Anal. 16, 179–199 (1996)MathSciNetCrossRef
16.
Franco, J.M.: Exponentially fitted symplectic integrators of RKN type for solving oscillatory problems. Comput. Phys. Commun. 177, 479–492 (2007)MathSciNetCrossRef
17.
French, D.A., Schaeffer, J.W.: Continuous finite element methods which preserve energy properties for nonlinear problems. Appl. Math. Comput. 39, 271–295 (1990)MathSciNetMATH
18.
Gautschi, W.: Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. Math. 3, 381–397 (1961)MathSciNetCrossRef
19.
20.
21.
Hairer, E.: Energy-preserving variant of collocation methods. J. Numer. Anal. Ind. Appl. Math. 5, 73–84 (2010)MathSciNetMATH
22.
Hairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2000)MathSciNetCrossRef
23.
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, 2nd edn. Springer, Berlin (2006)MATH
24.
Huang, N.S., Sidge, R.B., Cong, N.H.: On functionally fitted Runge-Kutta methods. BIT Numer. Math. 46, 861–874 (2006)MathSciNetCrossRef
25.
Hulme, B.L.: One-step piecewise polynomial Galerkin methods for initial value problems. Math. Comput. 26, 415–426 (1972)MathSciNetCrossRef
26.
Iserles, A.: On the method of Neumann series for highly oscillatory equations. BIT Numer. Math. 44, 473–488 (2004)MathSciNetCrossRef
27.
Ixaru, L.G., Vanden Bergehe, G. (eds.): Exponential Fitting. Kluwer Academic Publishers, Dordrecht (2004)MATH
28.
Li, Y.W., Wu, X.Y.: Functionally-fitted energy-preserving methods for solving oscillatory nonlinear Hamiltonian systems. SIAM J. Numer. Anal. 54, 2036–2059 (2016)MathSciNetCrossRef
29.
Mclachlan, R.I., Quispel, G.R.W., Robidoux, N.: Geometric integration using dicrete gradients. Philos. Trans. R. Soc. A 357, 1021–1046 (1999)CrossRef
30.
Miyatake, Y.: An energy-preserving exponentially-fitted continuous stage Runge-Kutta method for Hamiltonian systems. BIT Numer. Math. 54, 777–799 (2014)MathSciNetCrossRef
31.
Miyatake, Y.: A derivation of energy-preserving exponentially-fitted integrators for poisson systems. Comput. Phys. Commun. 187, 156–161 (2015)MathSciNetCrossRef
32.
Ozawa, K.: A functionally fitting Runge-Kutta method with variable coefficients. Jpn. J. Ind. Appl. Math. 18, 107–130 (2001)CrossRef
33.
Peregrine, D.H.: Water waves, nonlinear Schrödinger equations and their solutions. J. Austral. Math. Soc. Ser B 25, 16–43 (1983)MathSciNetCrossRef
34.
Petzold, L.R., Jay, L.O., Jeng, Y.: Numerical solution of highly oscillatory ordinary differential equations. Acta Numer. 6, 437–483 (1997)MathSciNetCrossRef
35.
Simos, J.C.: Assessment of energy-momentum and symplectic schemes for stiff dynamical systems. In: American Sociery for Mechanical Engineers, ASME Winter Annual meeting, New Orleans, Louisiana (1993)
36.
Simos, T.E.: Does variable step size ruin a symplectic integrator? Phys. D. Nonlinear Phenom. 60, 311–313 (1992)MathSciNetCrossRef
37.
Simos, T.E.: An exponentially-fitted Rung-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions. Comput. Phys. Commun. 115, 1–8 (1998)CrossRef
38.
Tang, W., Sun, Y.: Time finite element methods : A unified framework for the numerical discretizations of ODEs. Appl. Math. Comput. 219, 2158–2179 (2012)MathSciNetMATH
39.
Vande Vyver, H.: A fourth order symplectic exponentially fitted integrator. Comput. Phys. Commun. 176, 255–262 (2006)MathSciNetCrossRef
40.
Vanden Berghe, G., Daele, M., Vande Vyver, H.: Exponentially-fitted Runge-Kutta methods of collocation type : fixed or variable knots? J. Comput. Appl. Math. 159, 217–239 (2003)MathSciNetCrossRef
41.
Wang, B., Wu, X.Y.: A new high precision energy-preserving integrator for system of second-order differential equations. Phys. Lett. A 376, 1185–1190 (2012)MathSciNetCrossRef
42.
Wang, B., Iserles, A., Wu, X.Y.: Arbitrary order trigonometric fourier collocation methods for multi-frequency oscillatory systems. Found. Comput. Math. 16, 151–181 (2016)MathSciNetCrossRef
43.
Wu, X.Y., Wang, B., Xia, J.: Explicit symplectic multidimensional exponential fitting modified Runge-Kutta-Nyström methods. BIT Numer. Math. 52, 773–791 (2012)CrossRef
44.
Wu, X.Y., You, X., Wang, B.: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer, Berlin (2013)CrossRef
45.
Wu, X.Y., Liu, K., Shi, W.: Structure-Preserving Algorithms for Oscillatory Differential Equations II. Springer, Berlin (2015)CrossRef
46.
Yang, H., Wu, X.Y., You, X., Fang, Y.: Extended RKN-type methods for numerical integration of perturbed oscillators. Comput. Phys. Commun. 180, 1777–1794 (2009)MathSciNetCrossRef
Metadaten
Titel
Functionally Fitted Continuous Finite Element Methods for Oscillatory Hamiltonian Systems
verfasst von
Xinyuan Wu
Bin Wang
Copyright-Jahr
2018
Verlag
Springer Singapore
Buch
Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations

Print ISBN: 978-981-10-9003-5

Electronic ISBN: 978-981-10-9004-2

Copyright-Jahr: 2018

DOI
https://doi.org/10.1007/978-981-10-9004-2_1