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Exponential Average-Vector-Field Integrator for Conservative or Dissipative Systems

  • Xinyuan Wu
  • Bin Wang
Chapter

Abstract

This chapter focuses on discrete gradient integrators intending to preserve the first integral or the Lyapunov function of the original continuous system. Incorporating the discrete gradients with exponential integrators, we discuss a novel exponential integrator for the conservative or dissipative system \(\dot{y}=Q(My+\nabla U(y))\), where Q is a \(d\times d\) real matrix, M is a \(d\times d\) symmetric real matrix and \(U : \mathbb {R}^{d}\rightarrow \mathbb {R}\) is a differentiable function. For conservative systems, the exponential integrator preserves the energy, while for dissipative systems, the exponential integrator preserves the decaying property of the Lyapunov function. Two properties of the new scheme are presented. Numerical experiments demonstrate the remarkable superiority of the new scheme in comparison with other structure-preserving schemes in the recent literature.

References

  1. 1.
    Berland, H., Owren, B., Skaflestad, B.: Solving the nonlinear Schrödinger equations using exponential integrators on the cubic Schrödinger equation. Model. Identif. Control 27, 201–217 (2006)CrossRefGoogle Scholar
  2. 2.
    Berland, H., Skaflestad, B., Wright, W.: EXPINT – A MATLAB package for exponential integrators, ACM Trans. Math. Softw. 33 (2007)CrossRefGoogle Scholar
  3. 3.
    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)MathSciNetMATHGoogle Scholar
  4. 4.
    Calvo, M., Laburta, M.P., Montijano, J.I., Rández, L.: Projection methods preserving Lyapunov functions. BIT Numer. Math. 50, 223–241 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Celledoni, E., Cohen, D., Owren, B.: Symmetric exponential integrators with an application to the cubic Schrödinger equation. Found. Comp. Math. 8, 303–317 (2008)CrossRefGoogle Scholar
  6. 6.
    Celledoni, E., Grimm, V., McLachlan, R.I., Maclaren, 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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cieśliński, J.L.: Locally exact modifications of numerical schemes. Comput. Math. Appl. 62, 1920–1938 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cohen, D.: Conservation properties of numerical integrators for highly oscillatory Hamiltonian systems. IMA J. Numer. Anal. 26, 34–59 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    Cox, S.M., Matthews, P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Deuflhard, P.: A study of extrapolation methods based on multistep schemes without parasitic solutions. Z. Angew. Math. Phys. 30, 177–189 (1979)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Franco, J.: Runge–Kutta–Nyström methods adapted to the numerical integration of perturbed oscillators. Comput. Phys. Commun. 147, 770–787 (2002)CrossRefGoogle Scholar
  13. 13.
    Furihata, D., Matuso, T.: Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations. Chapman and Hall/CRC, Boca Raton (2010)CrossRefGoogle Scholar
  14. 14.
    Gautschi, W.: Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. Math. 3, 381–397 (1961)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gonzalez, O.: Time integration and discrete Hamiltonian systems. J. Nonlinear Sci. 6, 449–467 (1996)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)CrossRefGoogle Scholar
  17. 17.
    Hairer, E.: Energy-preserving variant of collocation methods. J. Numer. Anal. Ind. Appl. Math. 5, 73–84 (2010)MathSciNetMATHGoogle Scholar
  18. 18.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, vol. XIII, 2nd edn. Springer, Berlin (2006)MATHGoogle Scholar
  19. 19.
    Hernández-Solano, Y., Atencia, M., Joya, G., Sandoval, F.: A discrete gradient method to enhance the numerical behavior of Hopfield networks. Neurocomputing 164, 45–55 (2015)CrossRefGoogle Scholar
  20. 20.
    Hersch, J.: Contribution à la méthode des équations aux différences. Z. Angew. Math. Phys. 9, 129–180 (1958)CrossRefGoogle Scholar
  21. 21.
    Higham, N.J.: Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008)CrossRefGoogle Scholar
  22. 22.
    Hochbruck, M., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19, 1552–1574 (1998)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numerica 19, 209–286 (2010)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kassam, A.K., Trefethen, L.N.: Fourth order time-stepping for stiff PDEs. SIAM. J. Sci. Comput. 26, 1214–1233 (2005)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Klein, C.: Fourth order time-stepping for low dispersion Korteweg–de Vries and nonlinear Schrödinger equations. Electron. Trans. Numer. Anal. 29, 116–135 (2008)MATHGoogle Scholar
  26. 26.
    Lawson, J.D.: Generalized Runge–Kutta processes for stable systems with large Lipschitz constants. SIAM. J. Numer. Anal. Model. 4, 372–380 (1967)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Li, Y.W., Wu, X.Y.: Exponential integrators preserving first integrals or Lyapunov functions for conservative or dissipative systems. SIAM J. SCI. Comput. 38, A1876–A1895 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Macías-Díaz, J.E., Medina-Ramírez, I.E.: An implicit four-step computational method in the study on the effects of damping in a modified \(\alpha \)-Fermi-Pasta-Ulam medium. Commun. Nonlinear Sci. Numer. Simul. 14, 3200–3212 (2009)MathSciNetCrossRefGoogle Scholar
  29. 29.
    McLachlan, R.I., Quispel, G.R.W., Robidoux, N.: A unified approach to Hamiltonian systems, Poisson systems, gradient systems, and systems with Lyapunov functions or first integrals. Phys. Rev. Lett. 81, 2399–2411 (1998)MathSciNetCrossRefGoogle Scholar
  30. 30.
    McLachlan, R.I., Quispel, G.R.W., Robidoux, N.: Geometric integration using dicrete gradients. Philos. Trans. R. Soc. A 357, 1021–1046 (1999)CrossRefGoogle Scholar
  31. 31.
    Pavlov, B.V., Rodionova, O.E.: The method of local linearization in the numerical solution of stiff systems of ordinary differential equations. USSR Comput. Math. Math. Phys. 27, 30–38 (1987)CrossRefGoogle Scholar
  32. 32.
    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)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Wu, X.Y., Wang, B., Xia, J.: Explicit symplectic multidimensional exponential fitting modified Runge–Kutta–Nyström methods. BIT Numer. Math. 52, 773–795 (2012)CrossRefGoogle Scholar
  34. 34.
    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)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. And Science Press 2018

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingChina
  2. 2.School of Mathematical SciencesQufu Normal UniversityQufuChina

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