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
BIT Numerical Mathematics
Erschienen in: BIT Numerical Mathematics 3/2014

01.09.2014

An energy-preserving exponentially-fitted continuous stage Runge–Kutta method for Hamiltonian systems

verfasst von: Yuto Miyatake

Erschienen in: BIT Numerical Mathematics | Ausgabe 3/2014

Einloggen

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

search-config
loading …

Abstract

Recently, the symplectic exponentially-fitted methods for Hamiltonian systems with periodic or oscillatory solutions have been attracting a lot of interest. As an alternative to them, in this paper, we propose a class of energy-preserving exponentially-fitted methods. For this aim, we show sufficient conditions for energy-preservation in terms of the coefficients of continuous stage Runge–Kutta (RK) methods, and extend the theory of exponentially-fitted RK methods in the context of continuous stage RK methods. Then by combining these two theories, we derive second and fourth order energy-preserving exponentially-fitted schemes.
Vorheriger Artikel On an exact numerical simulation of solitary-wave solutions of the Burgers–Huxley equation through Cardano’s method
Nächster Artikel A weak second-order split-step method for numerical simulations of stochastic differential equations

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

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

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+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
Literatur
1.
Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)CrossRef
2.
3.
Ascher, U.M., Reich, S.: On some difficulties in integrating highly oscillatory Hamiltonian systems, pp. 281–296. In: Computational Molecular Dynamics. Lecture Notes in Computational Science and Engineering, vol. 4. Springer, Berlin (1999)
4.
Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comp. Math. Appl. Mech. Eng. 1, 1–16 (1972)MathSciNetCrossRefMATH
5.
6.
Calvo, M., Franco, J.M., Montijano, J.I., Rández, L.: Sixth-order symmetric and symplectic exponentially fitted modified Runge–Kutta methods of Gauss type. Comput. Phys. Comm. 178, 732–744 (2008)MathSciNetCrossRefMATH
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)MathSciNetCrossRefMATH
8.
Calvo, M., Franco, J.M., Montijano, J.I., Rández, L.: Sixth-order symmetric and symplectic exponentially fitted Runge–Kutta methods of the Gauss type. J. Comput. Appl. Math. 223, 387–398 (2009)MathSciNetCrossRefMATH
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 50, 3–21 (2010)MathSciNetCrossRefMATH
10.
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)CrossRefMATH
11.
Celledoni, E., McLachlan, R.I., McLaren, D.I., Owren, B., Quispel, G.R.W., Wright, W.M.: Energy-preserving Runge–Kutta methods. ESAIM Math. Model. Numer. Anal. 43, 645–649 (2009)MathSciNetCrossRefMATH
12.
Celledoni, E., McLachlan, R.I., Owren, B., Quispel, G.R.W.: Energy-preserving integrators and the structure of B-series. Tech. rep., NTNU preprint series: Numerics No.5/2009
13.
Chartier, P., Faou, E., Murua, A.: An algebraic approach to invariant preserving integators: the case of quadratic and Hamiltonian invariants. Numer. Math. 103, 575–590 (2006)MathSciNetCrossRefMATH
14.
15.
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
16.
Dahlby, M., Owren, B., Yaguchi, T.: Preserving multiple first integrals by discrete gradients. J. Phys. A 44, 305, 205 (2011)
17.
Deuflhard, P.: A study of extrapolation methods based on multistep schemes without parasitic solutions. Z Angew. Math. Phys. 30, 177–189 (1979)MathSciNetCrossRefMATH
18.
Eich, E.: Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints. SIAM J. Num. Anal. 30, 1467–1482 (1993)MathSciNetCrossRefMATH
19.
Franco, J.M.: An embedded pair of exponentially fitted explicit Runge–Kutta methods. J. Comput. Appl. Math. 149, 407–414 (2002)MathSciNetCrossRefMATH
20.
21.
Franco, J.M.: Runge-Kutta methods adapted to the numerical integration of oscillatory problems. Appl. Numer. Math. 50, 427–443 (2004)MathSciNetCrossRefMATH
22.
Franco, J.M.: Exponentially fitted symplectic integrators of RKN type for solving oscillatory problems. Comput. Phys. Commun. 177, 479–492 (2007)MathSciNetCrossRefMATH
23.
Garcia-Archilla, B., Sanz-Serna, J.M., Skeel, R.D.: Long-time-step methods for oscillatory differential equations. SIAM J. Sci. Comput. 20, 930–963 (1998)MathSciNetCrossRef
24.
Gautschi, W.: Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. Math. 3, 381–397 (1961)MathSciNetCrossRefMATH
25.
26.
Grimm, V., Hochbruck, M.: Error analysis of exponential integrators for oscillatory second-order differential equations. J. Phys. A 39, 5495 (2006)MathSciNetCrossRefMATH
27.
28.
29.
Hairer, E.: Energy-preserving variant of collocation methods. JNAIAM 5, 73–84 (2010)MathSciNet
30.
Hairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2000)MathSciNetCrossRefMATH
31.
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Heidelberg (2006)
32.
Hairer, E., McLachlan, R.I., Razakarivony, A.: Achieving Brouwer’s law with implicit Runge–Kutta methods. BIT 48, 231–243 (2008)MathSciNetCrossRefMATH
33.
Hairer, E., Zbinden, C.J.: On conjugate symplecticity of B-seried integrator. IMA J. Numer. Anal. 33, 57–79 (2013)
34.
Hochbruck, M., Lubich, C.: A Gautschi-type method for oscillatory second-order differential equations. Numer. Math. 83, 403–426 (1999)MathSciNetCrossRefMATH
35.
36.
Leimkuhler, B., Reich, S.: Simulating Hamiltonian Dynamics. Cambridge University Press, Cambridge (2004)MATH
37.
38.
Ozawa, K.: A functional fitting Runge–Kutta method with variable coefficients. Japan J. Indust. Appl. Math. 18, 107–130 (2001)MathSciNetCrossRefMATH
39.
Ozawa, K.: A functionally fitted three-stage explicit singly diagonally implicit Runge–Kutta method. Japan J. Indust. Appl. Math. 22, 403–427 (2005)MathSciNetCrossRefMATH
40.
Paternoster, B.: Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials. Appl. Numer. Math. 28, 401–412 (1998)MathSciNetCrossRefMATH
41.
Quispel, G.R.W., McLaren, D.I.: A new class of energy-preserving numerical integration methods. J. Phys. A 41, 045, 206 (2008)
42.
Quispel, G.R.W., Turner, G.S.: Discrete gradient methods for solving ODEs numerically while preserving a first integral. J. Phys. A 29, L341–L349 (1996)MathSciNetCrossRefMATH
43.
44.
Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994)CrossRefMATH
45.
Simos, T.E.: An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions. Comput. Phys. Commun. 115, 1–8 (1998)MathSciNetCrossRefMATH
46.
Suris, Y.B.: On the conservation of the symplectic structure in the numerical solution of Hamiltonian systems (in Russian). In: Filippov, S.S. (ed.) Numerical Solution of Ordinary Differential Equations, pp. 148–160. Keldysh Institute of Applied Mathematics, USSR Academy of Sciences, Moscow (1988)
47.
Tang, W., Sun, Y.: Time finite element methods: a unified framework for numerical discretizations of ODEs. Appl. Math. Comput. 219, 2158–2179 (2012)MathSciNetCrossRefMATH
48.
Van Daele, M., Vanden Berghe, G.: Geometric numerical integration by means of exponentially-fitted methods. Appl. Numer. Math. 57, 415–435 (2007)MathSciNetCrossRefMATH
49.
Van de Vyver, H.: A symplectic exponentially fitted modified Runge–Kutta-Nyström method for the numerical integration of orbital problems. New Astron. 10, 261–269 (2005)CrossRef
50.
Van de Vyver, H.: A fourth-order symplectic exponentially fitted integrator. Comput. Phys. Commun. 174, 255–262 (2006)CrossRefMATH
51.
Vanden Berghe, G., De Meyer, H., Van Daele, M., Van Hecke, T.: Exponentially-fitted explicit Runge-Kutta methods. Comput. Phys. Commun. 123, 7–15 (1999)
52.
Vanden Berghe, G., De Meyer, H., Van Daele, M., Van Hecke, T.: Exponentially fitted Runge-Kutta methods. J. Comput. Appl. Math. 125, 107–115 (2000)
53.
Vanden Berghe, G., Ixaru, L.G., De Meyer, H.: Frequency determination and step-length control for exponentially-fitted Runge-Kutta methods. J. Comput. Appl. Math. 132, 95–105 (2001)
54.
Vanden Berghe, G., Van Daele, M.: Symplectic exponentially-fitted four-stage Runge-Kutta methods of the Gauss type. Numer. Algor. 56, 591–608 (2011)
55.
Vanden Berghe, G., Van Daele, M., Vande Vyver, H.: Exponential fitted Runge–Kutta methods of collocation type: fixed or variable knot points? J. Comput. Appl. Math. 159, 217–239 (2003)
56.
Wu, X., Wang, B., Shi, W.: Efficient energy-preserving integrators for oscillatory Hamiltonian systems. J. Comput. Phys. 235, 587–605 (2013)MathSciNetCrossRefMATH
57.
Zhong, G., Marsden, J.E.: Lie-Poisson Hamilton–Jacobi theory and Lie–Poisson integrators. Phys. Lett. A 133, 134–139 (1988)MathSciNetCrossRef
Metadaten
Titel
An energy-preserving exponentially-fitted continuous stage Runge–Kutta method for Hamiltonian systems
verfasst von
Yuto Miyatake
Publikationsdatum
01.09.2014
Verlag
Springer Netherlands
Erschienen in
BIT Numerical Mathematics / Ausgabe 3/2014
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-014-0474-4

Weitere Artikel der Ausgabe 3/2014

BIT Numerical Mathematics 3/2014 Zur Ausgabe

OriginalPaper

A splitting preconditioner for the iterative solution of implicit Runge-Kutta and boundary value methods

OriginalPaper

On an exact numerical simulation of solitary-wave solutions of the Burgers–Huxley equation through Cardano’s method

Preface

Preface to 54-3

OriginalPaper

Second-order LOD multigrid method for multidimensional Riesz fractional diffusion equation

OriginalPaper

Raising the approximation order of multivariate quasi-interpolants

OriginalPaper

Accurate solutions of diagonally dominant tridiagonal linear systems

Premium Partner

    Bildnachweise