Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Functionally Fitted Continuous Finite Element Methods for Oscillatory Hamiltonian Systems Read chapter Read first chapter

2018 | OriginalPaper | Chapter

3. Exponential Fourier Collocation Methods for First-Order Differential Equations

Authors : Xinyuan Wu, Bin Wang

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

Publisher: Springer Singapore

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Commencing from the variation-of-constants formula and incorporating a local Fourier expansion of the underlying problem with collocation methods, this chapter presents a novel class of exponential Fourier collocation methods (EFCMs) for solving systems of first-order ordinary differential equations. We discuss in detail the connections of EFCMs with trigonometric Fourier collocation methods (TFCMs), the well-known Hamiltonian Boundary Value Methods (HBVMs), Gauss methods and Radau IIA methods. It turns out that the novel EFCMs are an extension, in a strict mathematical sense, of these existing methods in the literature.

Dont have a licence yet? Then find out more about our products and how to get one now:

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!

inform now

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!

inform now

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!

inform now
previous chapter Exponential Average-Vector-Field Integrator for Conservative or Dissipative Systems
next chapter Symplectic Exponential Runge–Kutta Methods for Solving Nonlinear Hamiltonian Systems
Literature
1.
Al-Mohy, A.H., Higham, N.J.: A new scaling and squaring algorithm for the matrix exponential. SIAM J. Matrix Anal. Appl. 31, 970–989 (2009)MathSciNetCrossRef
2.
Al-Mohy, A.H., Higham, N.J.: Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33, 488–511 (2011)MathSciNetCrossRef
3.
Berland, H., Owren, B., Skaflestad, B.: B-series and order conditions for exponential integrators. SIAM J. Numer. Anal. 43, 1715–1727 (2005)MathSciNetCrossRef
4.
Berland, H., Skaflestad, B., Wright, W.M.: EXPINT–A MATLAB package for exponential integrators. ACM Trans. Math. Softw. (TOMS) 33, 4 (2007)CrossRef
5.
Brugnano, L., Iavernaro, F., Magherini, C.: Efficient implementation of Radau collocation methods. Appl. Numer. Math. 87, 100–113 (2015)MathSciNetCrossRef
6.
Brugnano, L., Iavernaro, F., Trigiante, D.: Hamiltonian boundary value methods (energy preserving discrete line integral methods). J. Numer. Anal. Ind. Appl. Math. 5, 17–37 (2010)MathSciNetMATH
7.
Brugnano, L., Iavernaro, F., Trigiante, D.: A note on the efficient implementation of Hamiltonian BVMs. J. Comput. Appl. Math. 236, 375–383 (2011)MathSciNetCrossRef
8.
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
9.
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
10.
Brugnano, L., Mazzia, F., Trigiante, D.: Fifty years of stiffness, Recent Advances in Computational and Applied Mathematics, pp. 1–21. Springer, The Netherlands (2011)CrossRef
11.
Caliari, M., Ostermann, A.: Implementation of exponential Rosenbrock-type integrators. Appl. Numer. Math. 59, 568–581 (2009)MathSciNetCrossRef
12.
Calvo, M.P., Palencia, C.: A class of explicit multistep exponential integrators for semilinear problems. Numer. Math. 102, 367–381 (2006)MathSciNetCrossRef
13.
Celledoni, E., Cohen, D., Owren, B.: Symmetric exponential integrators with an application to the cubic Schrödinger equation. Found. Comput. Math. 8, 303–317 (2008)MathSciNetCrossRef
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.
16.
Grimm, V., Hochbruck, M.: Error analysis of exponential integrators for oscillatory second-order differential equations. J. Phys. A Math. Gen. 39, 5495–5507 (2006)MathSciNetCrossRef
17.
Hairer, E.: Energy-preserving variant of collocation methods, JNAIAM J. Numer. Anal. Ind. Appl. Math. 5, 73–84 (2010)MathSciNetMATH
18.
Hairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2000)MathSciNetCrossRef
19.
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin (2006)MATH
20.
Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996)CrossRef
21.
Hale, J.K.: In: Roberte, E. (ed.) Ordinary Differential Equations. Krieger Publishing Company, Huntington (1980)
22.
23.
Hochbruck, M., Lubich, C.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911–1925 (1997)MathSciNetCrossRef
24.
Hochbruck, M., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19, 1552–1574 (1998)MathSciNetCrossRef
25.
Hochbruck, M., Ostermann, A.: Explicit exponential Runge–Kutta methods for semilineal parabolic problems. SIAM J. Numer. Anal. 43, 1069–1090 (2005)MathSciNetCrossRef
26.
Hochbruck, M., Ostermann, A.: Exponential Runge–Kutta methods for parabolic problems. Appl. Numer. Math. 53, 323–339 (2005)MathSciNetCrossRef
27.
28.
Hochbruck, M., Ostermann, A., Schweitzer, J.: Exponential rosenbrock-type methods. SIAM J. Numer. Anal. 47, 786–803 (2009)MathSciNetCrossRef
29.
Iavernaro, F., Trigiante, D.: High-order symmetric schemes for the energy conservation of polynomial Hamiltonian problems. JNAIAM J. Numer. Anal. Ind. Appl. Math. 4, 101–787 (2009)MathSciNetMATH
30.
Iserles, A.: On the global error of discretization methods for highly-oscillatory ordinary differential equations. BIT 42, 561–599 (2002)MathSciNetCrossRef
31.
Iserles, A.: Think globally, act locally: solving highly-oscillatory ordinary differential equations. Appl. Num. Anal. 43, 145–160 (2002)MathSciNetMATH
32.
Kassam, A.K., Trefethen, L.N.: Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26, 1214–1233 (2005)MathSciNetCrossRef
33.
Khanamiryan, M.: Quadrature methods for highly oscillatory linear and nonlinear systems of ordinary differential equations: part I. BIT Num. Math. 48, 743–762 (2008)MathSciNetCrossRef
34.
35.
Lubich, C.: From quantum to classical molecular dynamics: reduced models and numerical analysis, European Mathematical Society (2008)
36.
Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45, 3–49 (2003)MathSciNetCrossRef
37.
Ostermann, A., Thalhammer, M., Wright, W.M.: A class of explicit exponential general linear methods. BIT Numer. Math. 46, 409–431 (2006)MathSciNetCrossRef
38.
39.
Wang, B., Iserles, A.: Dirichlet series for dynamical systems of first-order ordinary differential equations. Discret. Contin. Dyn. Syst. B 19, 281–298 (2014)MathSciNetCrossRef
40.
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
41.
Wang, B., Li, G.: Bounds on asymptotic-numerical solvers for ordinary differential equations with extrinsic oscillation. Appl. Math. Modell. 39, 2528–2538 (2015)MathSciNetCrossRef
42.
Wang, B., Liu, K., Wu, X.Y.: A Filon-type asymptotic approach to solving highly oscillatory second-order initial value problems. J. Comput. Phys. 243, 210–223 (2013)MathSciNetCrossRef
43.
Wang, B., Wu, X.Y.: A new high precision energy-preserving integrator for system of oscillatory second-order differential equations. Phys. Lett. A 376, 1185–1190 (2012)MathSciNetCrossRef
44.
Wang, B., Wu, X.Y., Meng, F.: Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second-order differential equations. J. Comput. Appl. Math. 313, 185–201 (2017)MathSciNetCrossRef
45.
Wang, B., Wu, X.Y., Meng, F.: Exponential Fourier collocation methods for first-order differential equations. J. Comput. Math. 35, 711–736 (2017)MathSciNetCrossRef
46.
Wu, X.Y., Wang, B., Shi, W.: Efficient energy-preserving integrators for oscillatory Hamiltonian systems. J. Comput. Phys. 235, 587–605 (2013)MathSciNetCrossRef
47.
Wu, X.Y., Wang, B., Xia, J.: Explicit symplectic multidimensional exponential fitting modified Runge–Kutta–Nyström methods. BIT 52, 773–795 (2012)MathSciNetCrossRef
48.
Wu, X.Y., You, X., Wang, B.: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer, Berlin (2013)CrossRef
Metadata
Title
Exponential Fourier Collocation Methods for First-Order Differential Equations
Authors
Xinyuan Wu
Bin Wang
Copyright Year
2018
Publisher
Springer Singapore
Book
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 Year: 2018

DOI
https://doi.org/10.1007/978-981-10-9004-2_3

Premium Partner

    Image Credits