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Geometric Integration Methods that Preserve Lyapunov Functions

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Abstract

We consider ordinary differential equations (ODEs) with a known Lyapunov function V. To ensure that a numerical integrator reflects the correct dynamical behaviour of the system, the numerical integrator should have V as a discrete Lyapunov function. Only second-order geometric integrators of this type are known for arbitrary Lyapunov functions. In this paper we describe projection-based methods of arbitrary order that preserve any given Lyapunov function.

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References

  1. U. Ascher and S. Reich, On some difficulties in integrating highly oscillatory Hamiltonian systems, in Computational Molecular Dynamics, Lect. Notes Comput. Sci. Eng. 4, Springer, Berlin, 1999, pp. 281–296.

  2. C. J. Budd and A. Iserles, eds., Geometric Integration: Numerical solution of differential equations on manifolds, Special edition of Philos. Trans. R. Soc. Lond., 357 (1999).

  3. C. J. Budd and M. D. Piggott, Geometric integration and its applications, Proceedings of ECMWF workshop on “Developments in numerical methods for very high resolution global models”, 2000, pp. 93–118.

  4. C. M. Elliot and A. M. Stuart, The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal., 30 (1993), pp. 1622–1663.

    Google Scholar 

  5. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.

  6. E. Hairer, A. Iserles and J. M. Sanz-Serna, Equilibria of Runge-Kutta methods, Numer. Math., 38 (1990), pp. 243–254.

  7. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer, Berlin, Heidelberg, 1996.

  8. E. Hairer, Symmetric Projection Methods for Differential Equations on Manifolds, BIT, 40 (2000), pp. 726–734.

  9. E. Hairer, Ch. Lubich and G. Wanner, Geometric Numerical Integration, Springer, 2002.

  10. K. Heun, Neue Methode zur approximativen Integration der Differentialgleichungen einer unabhängigen Veränderlichen, Z. Math. Phys., 45 (1900), pp. 23–38.

  11. A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods, Acta Numer., 9 (2000), pp. 215–365.

  12. V. Lakshmikantham, V. M. Matrosov and S. Sivasundaram, Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems, Kluwer, Dordrecht, 1991.

  13. B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, Cambridge University Press, 2004.

  14. A. A. Martynyuk, Stability by Liapunov’s Matrix Function Method with Applications, Marcel Dekker, New York, Basel, Hong Kong, 1998.

  15. R. I. McLachlan, G. R. W. Quispel and N. Robidoux, A unified approach to Hamiltonian systems, Poisson systems, gradient systems, and systems with a Lyapunov function and/or first integrals, Phys. Rev. Lett., 81 (1998), pp. 2399–2403.

    Google Scholar 

  16. R. I. McLachlan, G. R. W. Quispel and N. Robidoux, Geometric integration using discrete gradients, Philos. Trans. R. Soc. Lond. A, 357 (1999), pp. 1021–1045.

  17. R. I. McLachlan and G. R. W. Quispel, Six lectures on the geometric integration of ODEs, in Foundations of Computational Mathematics, C. U. P., R. A. DeVore et al., eds., 2001, pp. 155–210.

  18. R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numer., 11 (2002), pp. 341–434.

  19. J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Chapman & Hall, London, 1994.

  20. H. J. Stetter, Analysis of Discretisation Methods for Ordinary Differential Equations, Springer, Berlin, 1973.

  21. A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge Univ. Press, New York, 1996.

  22. G. Wanner, Runge-Kutta-methods with expansion in even powers of h, Computing, 11 (1973), pp. 81–85.

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Authors and Affiliations

  1. Department of Mathematics, and Centre for Mathematics and Statistics of Complex Systems, La Trobe University, Bundoora, Melbourne, 3086, Australia

    V. Grimm & G. R. W. Quispel

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  1. V. Grimm
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  2. G. R. W. Quispel
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Correspondence to V. Grimm.

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AMS subject classification (2000)

65L05, 65L06, 65L20, 65P40

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Grimm, V., Quispel, G. Geometric Integration Methods that Preserve Lyapunov Functions. Bit Numer Math 45, 709–723 (2005). https://doi.org/10.1007/s10543-005-0034-z

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