Abstract
The article considers symmetric general linear methods, a class of numerical time integration methods which, like symmetric Runge–Kutta methods, are applicable to general time-reversible differential equations, not just those derived from separable second-order problems. A definition of time-reversal symmetry is formulated for general linear methods, and criteria are found for the methods to be free of linear parasitism. It is shown that symmetric parasitism-free methods cannot be explicit, but such a method of order 4 is constructed with only one implicit stage. Several characterizations of symmetry are given, and connections are made with G-symplecticity. Symmetric methods are shown to be of even order, a suitable symmetric starting method is constructed and shown to be essentially unique. The underlying one-step method is shown to be time-symmetric.
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References
Butcher, J.C.: The equivalence of algebraic stability and \(AN\)-stability. BIT 27, 510–533 (1987)
Butcher, J.C., Habib, Y., Hill, A.T., Norton, T.J.T.: The control of parasitism in \(G\)-symplectic methods. SIAM J. Numer. Anal. 52, 2440–2465 (2014)
Cano, B., Sanz-Serna, J.M.: Error growth in the numerical integration of periodic orbits by multistep methods, with application to reversible systems. IMA J. Numer. Anal. 18, 57–75 (1998)
Cowell, P.H., Crommelin A.C.D.: Investigations in the motion of Halley’s comet from 1759 to 1910. Appendix to Greenwich Observations for 1909, Edinburgh, pp. 1–84 (1910)
D’Ambrosio, R., Hairer, E.: Long-term stability of multi-value methods for ordinary differential equations. J. Sci. Comput. 60, 627–640 (2014)
Dahlquist, G.: Convergence and stability in the numerical integration of ordinary differential equations. Math. Scand. 4, 33–53 (1956)
Dahlquist, G.: Stability and error bounds in the numerical integration of ordinary differential equations. Trans. Royal Inst. Techn., Stockholm, Sweden, vol. 130 (1959)
Eirola, T., Sanz-Serna, J.M.: Conservation of integrals and symplectic structure in the integration of differential equations by multistep methods. Numer. Math. 61, 281–290 (1992)
Faou, E., Hairer, E., Pham, T.-L.: Energy conservation with non-symplectic methods: examples and counter-examples. BIT 44, 699–709 (2004)
Hairer, E.: Symmetric linear multistep methods. BIT 46, 515–524 (2006)
Hairer, E., Lubich, C.: Symmetric multistep methods over long times. Numer. Math. 97, 699–723 (2004)
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration Structure-Preserving Algorithms for Ordinary Differential Equations. Spinger Verlag, Berlin (2002)
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Spinger Verlag, Berlin (2006)
Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations I, 2nd edn. Spinger Verlag, Berlin (1993)
Hairer, E., Stoffer, D.: Reversible long-term integration with variable step-sizes. SIAM J. Sci. Comput. 18, 257–269 (1997)
Hill, A.T.: Nonlinear stability of geneal linear methods. Numer. Math. 103, 611–629 (2006)
Hundsdorfer, W.H., Spijker, M.N.: A note on B-stability of Runge–Kutta methods. Numer. Math. 36, 319–331 (1981)
Kirchgraber, U.: Multi-step methods are essentially one-step methods. Numer. Math. 48, 85–90 (1986)
Lambert, J.D., Watson, I.A.: Symmetric multistep methods for periodic initial value problems. J. Inst. Math. Appl. 18, 189–202 (1976)
McLachlan, R.: On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comput. 16, 151–168 (1995)
Murua, A., Sanz-Serna, J.M.: Order conditiond for numerical integrators obtained by composing simpler integrators. Phil. Trans. Roy. Soc. A 357, 1079–1100 (1999)
Quinlan, G.D., Tremaine, S.: Symmetric multistep methods for the numerical integration of planetary orbits. Astron. J. 100, 1694–1700 (1990)
Sanz-Serna, J.M., Abia, L.: Order conditions for canonical Runge-Kutta schemes. SIAM J Numer. Anal. 28, 1081–1096 (1991)
Stetter, H.J.: Analysis of Discretization Methods for Ordinary Differential Equations. Springer Verlag, Berlin (1973)
Störmer, C.: Méthodes d’intégration numériquedes équations différentielles ordinaires. C.R. Congr. Intern. Math., Strasbourg, pp. 243–257 (1921)
Stoffer, D.: On reversible and canonical integration methods. Research Report No. 88-05 SAM. ETH, Zürich (1988)
Stoffer, D.: General linear methods: connection to one-step methods and invariant curves. Numer. Math. 64, 395–407 (1993)
Suzuki, M.: Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys. Lett. A 146, 319–323 (1990)
Verlet, L.: Computer ‘experiments’ on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev. 159, 98–103 (1967)
Wanner, G.: Runge-Kutta methods with expansion in even powers of \(h\). Computing 11, 81–85 (1973)
Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Clarendon Press, Oxford (1965)
Yoshida, H.: Construction of higher order symplectic integrators. Phys. Lett. A 150, 262–268 (1990)
Acknowledgments
JCB was supported by Marsden Grant AMC1101. ATH was assisted by LMS Grant 41125. TJTN was supported by a scholarship from EPSRC UK.
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Communicated by Christian Lubich.
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Butcher, J.C., Hill, A.T. & Norton, T.J.T. Symmetric general linear methods. Bit Numer Math 56, 1189–1212 (2016). https://doi.org/10.1007/s10543-016-0613-1
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DOI: https://doi.org/10.1007/s10543-016-0613-1