Hostname: page-component-7b9c58cd5d-wdhn8 Total loading time: 0 Render date: 2025-03-14T18:47:32.009Z Has data issue: false hasContentIssue false
  • English
  • Français

Symplectic integrators for Hamiltonian problems: an overview

Published online by Cambridge University Press:  07 November 2008

J. M. Sanz-Serna
J. M. Sanz-Serna
Affiliation:
Departamento de Matemática Aplicada y ComputaciónFacultad de CienciasUniversidad de Valladolid, Valladolid, Spain, E-mail: sanzserna@cpd.uva.es
Get access Rights & Permissions [Opens in a new window]

Extract

In the sciences, situations where dissipation is not significant may invariably be modelled by Hamiltonian systems of ordinary, or partial, differential equations. Symplectic integrators are numerical methods specifically aimed at advancing in time the solution of Hamiltonian systems. Roughly speaking, ‘symplecticness’ is a characteristic property possessed by the solutions of Hamiltonian problems. A numerical method is called symplectic if, when applied to Hamiltonian problems, it generates numerical solutions which inherit the property of symplecticness.

Type
Research Article
Information
Acta Numerica , Volume 1 , January 1992 , pp. 243 - 286
Copyright
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abia, L. and Sanz-Serna, J.M. (1990), ‘Partitioned Runge–Kutta methods for separable Hamiltonian problems’, Applied Mathematics and Computation Reports, Report 1990/8, Universidad de Valladolid.Google Scholar
Aizu, K. (1985), ‘Canonical transformation invariance and linear multistep formula for integration of Hamiltonian systems’, J. Comput. Phys. 58, 270–274.CrossRefGoogle Scholar
Anosov, D.V. and Arnold, V.I. (eds) (1988), Dynamical Systems I, Springer (Berlin).CrossRefGoogle Scholar
Arnold, V.I. (ed.) (1988), Dynamical Systems III, Springer (Berlin).CrossRefGoogle Scholar
Arnold, V.I. (1989), Mathematical Methods of Classical Mechanics, 2nd edition, Springer (New York).CrossRefGoogle Scholar
Arnold, V.I. and Novikov, S.P. (1990), Dynamical Systems IV, Springer, Berlin.CrossRefGoogle Scholar
Beyn, W.-J. (1991), ‘Numerical methods for dynamical systems’, in Advances in Numerical Analysis, Vol. I (Light, W., ed.) Clarendon Press (Oxford) 175–236.CrossRefGoogle Scholar
Burrage, K. and Butcher, J.C. (1979), ‘Stability criteria for implicit Runge-Kutta methods’, SIAM J. Numer. Anal. 16, 46–57.CrossRefGoogle Scholar
Butcher, J.C. (1976), ‘On the implementation of implicit Runge–Kutta methods’, BIT 16, 237–240.CrossRefGoogle Scholar
Butcher, J.C. (1987), The Numerical Analysis of Ordinary Differential Equations, John Wiley (Chichester).Google Scholar
Calvo, M.P. (1991), Ph.D. Thesis, University of Valladolid (to appear).Google Scholar
Calvo, M.P. and Sanz-Serna, J.M. (1991a), ‘Order conditions for canonical Runge–Kutta–Nyström methods’, BIT to appear.Google Scholar
Calvo, M.P. and Sanz-Serna, J.M. (1991b), ‘Variable steps for symplectic integrators’, Applied Mathematics and Computation Reports, Report 1991/3, Universidad de Valladolid.Google Scholar
Calvo, M.P. and Sanz-Serna, J.M. (1991c), ‘Reasons for a failure. The integration of the two-body problem with a symplectic Runge–Kutta–Nyström code with step-changing facilities’, Applied Mathematics and Computation Reports, Report 1991/7, Universidad de Valladolid.Google Scholar
Candy, J. and Rozmus, W. (1991), ‘A symplectic integration algorithm for separable Hamiltonian functions’, J. Comput. Phys. 92, 230–256.CrossRefGoogle Scholar
Channell, P. J., ‘Symplectic integration algorithms’, Los Alamos National Laboratory Report, Report AT-6ATN 83–9.Google Scholar
Channell, P.J. and Scovel, C. (1990), ‘Symplectic integration of Hamiltonian systems’, Nonlinearity 3, 231–259.CrossRefGoogle Scholar
Cooper, G.J. (1987), ‘Stability of Runge–Kutta methods for trajectory problems’, IMA J. Numer. Anal. 7, 1–13.CrossRefGoogle Scholar
Cooper, G.J. and Vignesvaran, R. (1990), ‘A scheme for the implementation of implicit Runge-Kutta methods’, Computing 45, 321–332.CrossRefGoogle Scholar
Crouzeix, M. (1979), ‘Sur la B-stabilité des méthodes de Runge–Kutta’, Numer. Math. 32, 75–82.CrossRefGoogle Scholar
Dekker, K. and Verwer, J.G. (1984), Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations, North Holland (Amsterdam).Google Scholar
Dormand, J.R., El-Mikkawy, M.E.A. and Prince, P.J. (1987), ‘Families of Runge–Kutta–Nyström formulae’, IMA J. Numer. Anal. 7, 235–250.CrossRefGoogle Scholar
Eirola, T. and Sanz-Serna, J.M. (1990), ‘Conservation of integrals and symplectic structure in the integration of differential equations by multistep methods’, Applied Mathematics and Computation Reports, Report 1990/9, Universidad de Valladolid.Google Scholar
Kang, Feng (1985), ‘On difference schemes and symplectic geometry’, in Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations (Kang, Feng, ed.) Science Press (Beijing) 42–58.Google Scholar
Kang, Feng (1986a), ‘Difference schemes for Hamiltonian formalism and symplectic geometry’, J. Comput. Math. 4, 279–289.Google Scholar
Kang, Feng (1986b), ‘Symplectic geometry and numerical methods in fluid dynamics’, in Tenth International Conference on Numerical Methods in Fluid Dynamics (Lecture Notes in Physics 264) (Zhuang, F. G. and Zhu, Y. L., eds) Springer (Berlin) 1–7.Google Scholar
Kang, Feng and Meng-zhao, Qin (1987), ‘The symplectic methods for the computation of Hamiltonian equations’, in Proceedings of the 1st Chinese Conference for Numerical Methods for PDE's, March 1986, Shanghai (Lecture Notes in Mathematics 1297) (You-lan, Zhu and Ben-yu, Gou, eds) Springer (Berlin) 1–37.Google Scholar
Kang, Feng, Hua-mo, Wu and Meng-zhao, Qin (1990), ‘Symplectic difference schemes for linear Hamiltonian canonical systems’, J. Comput. Math. 8, 371–380.Google Scholar
Kang, Feng, Hua-mo, Wu, Meng-zhao, Qin and Dao-liu, Wang (1989), ‘Construction of canonical difference schemes for Hamiltonian formalism via generating functions’, J. Comput. Math. 7, 71–96.Google Scholar
Forest, E. and Ruth, R. (1990), ‘Fourth-order symplectic integration’, Physica D 43, 105–117.CrossRefGoogle Scholar
de Frutos, J. and Sanz-Serna, J.M. (1991), ‘An easily implementable fourth-order method for the time integration of wave problems’, Applied Mathematics and Computation Reports, Report 1991/2, Universidad de Valladolid.Google Scholar
de Frutos, J., Ortega, T. and Sanz-Serna, J.M. (1990), ‘A Hamiltonian, explicit algorithm with spectral accuracy for the “good” Boussinesq system’, Comput. Methods Appl. Mech. Engrg. 80, 417–423.CrossRefGoogle Scholar
Zhong, Ge and Marsden, J.E. (1988), ‘Lie–Poisson Hamilton–Jacobi theory and Lie-Poisson integrators’, Phys. Lett. A 133, 134–139.CrossRefGoogle Scholar
Griffiths, D.F. and Sanz-Serna, J.M. (1986), ‘On the scope of the method of modified equations‘, SIAM J. Sci. Stat. Comput. 7, 994–1008.CrossRefGoogle Scholar
Guckenheimer, J. and Holmes, P. (1983), Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer (New York).CrossRefGoogle Scholar
Hairer, E., Iserles, A. and Sanz-Serna, J.M. (1990), ‘Equilibria of Runge–Kutta methods’, Numer. Math. 58, 243–254.CrossRefGoogle Scholar
Hairer, E., Nørsett, S.P. and Wanner, G. (1987), Solving Ordinary Differential Equations I, Nonstiff Problems, Springer (Berlin).CrossRefGoogle Scholar
Iserles, A. (1990a), ‘Stability and dynamics of numerical methods for ordinary differential equations’, IMA J. Numer. Anal. 10, 1–30.CrossRefGoogle Scholar
Iserles, A. (1990b), ‘Efficient Runge–Kutta methods for Hamiltonian equations’, Numerical Analysis Report DAMTP, Report 1990/NA10, University of Cambridge.Google Scholar
Iserles, A. and Nørsett, S.P. (1991), Order Stars, Chapman and Hall (London).CrossRefGoogle Scholar
Klein, F. (1926), Vorlesungen Über die Entwicklung der Mathematik im 19.Jahrhundert, Teil I, Springer (Berlin).Google Scholar
Lasagni, F. (1988), ‘Canonical Runge–Kutta methods’, ZAMP 39, 952–953.Google Scholar
Lasagni, F. (1990), ‘Integration methods for Hamiltonian differential equations’, unpublished manuscript.Google Scholar
Chun-wang, Li and Meng-zhao, Qin (1988), ‘A symplectic difference scheme for the infinite dimensional Hamiltonian system’, J. Comput. Math. 6, 164–174.Google Scholar
MacKay, R.S. (1991), ‘Some aspects of the dynamics and numerics of Hamiltonian systems’, in Proceedings of the ‘Dynamics of Numerics and Numerics of Dynamics’ Conference (Broomhead, D.S. and Iserles, A., eds) Oxford University Press (Oxford) to appear.Google Scholar
MacKay, R.S. and Meiss, J.D. (eds) (1987), Hamiltonian Dynamical Systems, Adam Hilger (Bristol).Google Scholar
McLachlan, R. and Atela, P. (1991), ‘The accuracy of symplectic integrators’, Program in Applied Mathematics, Report PAM 76, University of Colorado.Google Scholar
Menyuk, C.R. (1984), ‘Some properties of the discrete Hamiltonian method’, Physica D 11, 109–129.CrossRefGoogle Scholar
Miesbach, S. and Pesch, H.J. (1990), ‘Symplectic phase flow approximation for the numerical integration of canonical systems’, Schwerpunktprogramm der Deutschen Forschungsgemeinschaft, Anwendungsbezogene Optimierung und Steuerung, Report 233, Technische Universität München.Google Scholar
Moser, J. (1973), Stable and Random Motions in Dynamical Systems, Princeton University Press (Princeton).Google Scholar
Neishtadt, A.I. (1984), ‘The separation of motions in systems with rapidly rotating phase’, J. Appl. Math. Mech 48, 133–139.CrossRefGoogle Scholar
Neri, F. (1987), ‘Lie algebras and canonical integration’, University of Maryland Technical Report.Google Scholar
Okunbor, D. and Skeel, R.D. (1990), ‘An explicit Runge–Kutta–Nyström method is canonical if and only if its adjoint is explicit’, Working Document 90–3, Department of Computer Science, University of Illinois at Urbana-Champaign.Google Scholar
Okunbor, D. and Skeel, R.D. (1991), ‘Explicit canonical methods for Hamiltonian systems’, Working Document 91–1, Department of Computer Science, University of Illinois at Urbana-Champaign.Google Scholar
Pullin, D.I. and Saffman, P.G. (1991), ‘Long-tune symplectic integration: the example of four-vortex motion’, Proc. Roy. Soc. Lond. A 432, 481–494.Google Scholar
Meng-zhao, Qin (1988), ‘Leap-frog schemes of two kinds of Hamiltonian systems for wave equations’, Math. Num. Sin. 10, 272–281.Google Scholar
Meng-zhao, Qin and Mei-qing, Zhang (1990), ‘Multi-stage symplectic schemes of two kinds of Hamiltonian systems for wave equations’, Comput. Math. Appl. 19, 51–62.CrossRefGoogle Scholar
Ruth, R.D. (1983), ‘A canonical integration techniqe’, IEEE Trans. Nucl. Sci. 30, 2669–2671.CrossRefGoogle Scholar
Sanz-Serna, J.M. (1988), ‘Runge–Kutta schemes for Hamiltonian systems’, BIT 28, 877–883.CrossRefGoogle Scholar
Sanz-Serna, J.M. (1989) ‘The numerical integration of Hamiltonian systems’, in Proc. 1989 London Numerical ODE Conference, Oxford University Press (Oxford) to appear.Google Scholar
Sanz-Serna, J.M. (1990), ‘Numerical ordinary differential equations vs. dynamical systems’, Applied Mathematics and Computation Reports, Report 1990/3, Universidad de Valladolid.Google Scholar
Sanz-Serna, J.M. (1991a), ‘Two topics in nonlinear stability’, in Advances in Numerical Analysis, Vol. I (Light, W., ed.) Clarendon Press (Oxford) 147–174.CrossRefGoogle Scholar
Sanz-Serna, J.M. (1991b), ‘Symplectic Runge–Kutta and related methods: recent results’, Applied Mathematics and Computation Reports, Report 1991/5, Universidad de Valladolid.Google Scholar
Sanz-Serna, J.M. and Abia, L. (1991), ‘Order conditions for canonical Runge-Kutta schemes’, SIAM J. Numer. Anal. 28, 1081–1096.CrossRefGoogle Scholar
Sanz-Serna, J.M. and Griffiths, D.F. (1991), ‘A new class of results for the algebraic equations of implicit Runge-Kutta processes’, IMA J. Numer. Anal, to appear.CrossRefGoogle Scholar
Sanz-Serna, J.M. and Vadillo, F. (1986), ‘Nonlinear instability, the dynamic approach’, in Numerical Analysis (Griffiths, D.F. and Watson, G.A., eds) Longman (London) 187–199.Google Scholar
Sanz-Serna, J.M. and Vadillo, F. (1987), ‘Studies in nonlinear instability III: augmented Hamiltonian problems’, SIAM J. Appl. Math. 47, 92–108.CrossRefGoogle Scholar
Suris, Y.B. (1989), ‘Canonical transformations generated by methods of Runge-Kutta type for the numerical integration of the system x″ = −∂U∂x’ Zh. Vychisl. Mat. i Mat. Fiz. 29, 202–211 (in Russian). An English translation by D.V. Bobyshev, edited by R.D. Skeel, is available as Working Document 91–2, Numerical Computing Group, Department of Computer Science, University of Illinois at Urbana-Champaign.Google Scholar
Suris, Y.B. (1990), ‘Hamiltonian Runge–Kutta type methods and their variational formulation’, Math. Sim. 2, 78–87 (in Russian).Google Scholar
Warming, R.F. and Hyett, B.J. (1974), ‘The modified equation approach to the stability and accuracy analysis of finite difference methods’, J. Comput. Phys. 14, 159–179.CrossRefGoogle Scholar
Yoshida, H. (1990), ‘Construction of higher order symplectic integrators’, Phys. Lett. A 150, 262–268.CrossRefGoogle Scholar
Yuhua, Wu (1988), ‘The generating function for the solution of ODE's and its discrete methods’, Comput. Math. Appl. 15, 1041–1050.Google Scholar