Numerical integrators based on modified differential equations

Chartier, Philippe; Hairer, Ernst; Vilmart, Gilles

doi:10.1090/S0025-5718-07-01967-9

Numerical integrators based on modified differential equations
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by Philippe Chartier, Ernst Hairer and Gilles Vilmart;

Math. Comp. 76 (2007), 1941-1953

DOI: https://doi.org/10.1090/S0025-5718-07-01967-9

Published electronically: May 9, 2007

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Abstract:

Inspired by the theory of modified equations (backward error analysis), a new approach to high-order, structure-preserving numerical integrators for ordinary differential equations is developed. This approach is illustrated with the implicit midpoint rule applied to the full dynamics of the free rigid body. Special attention is paid to methods represented as B-series, for which explicit formulae for the modified differential equation are given. A new composition law on B-series, called substitution law, is presented.

References

P. J. Channell and C. Scovel, Symplectic integration of Hamiltonian systems, Nonlinearity 3 (1990), no. 2, 231–259. MR 1054575
P. Chartier, E. Hairer, and G. Vilmart. A substitution law for B-series vector fields. INRIA Report, No. 5498, 2005.
P. Chartier and E. Lapôtre. Reversible B-series. INRIA Report, No. 1221, 1998.
Kang Feng, Difference schemes for Hamiltonian formalism and symplectic geometry, J. Comput. Math. 4 (1986), no. 3, 279–289. MR 860157
Kang Feng, Hua Mo Wu, Meng Zhao Qin, and Dao Liu Wang, Construction of canonical difference schemes for Hamiltonian formalism via generating functions, J. Comput. Math. 7 (1989), no. 1, 71–96. MR 1017182
Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric numerical integration, 2nd ed., Springer Series in Computational Mathematics, vol. 31, Springer-Verlag, Berlin, 2006. Structure-preserving algorithms for ordinary differential equations. MR 2221614
Ernst Hairer and Gilles Vilmart, Preprocessed discrete Moser-Veselov algorithm for the full dynamics of a rigid body, J. Phys. A 39 (2006), no. 42, 13225–13235. MR 2266054, DOI 10.1088/0305-4470/39/42/003
E. Hairer and G. Wanner, On the Butcher group and general multi-value methods, Computing (Arch. Elektron. Rechnen) 13 (1974), no. 1, 1–15 (English, with German summary). MR 403225, DOI 10.1007/bf02268387
Benedict Leimkuhler and Sebastian Reich, Simulating Hamiltonian dynamics, Cambridge Monographs on Applied and Computational Mathematics, vol. 14, Cambridge University Press, Cambridge, 2004. MR 2132573
R. I. McLachlan. A new implementation of symplectic Runge–Kutta methods. To appear in SIAM J. Sci. Comput.
Robert I. McLachlan and Antonella Zanna, The discrete Moser-Veselov algorithm for the free rigid body, revisited, Found. Comput. Math. 5 (2005), no. 1, 87–123. MR 2125692, DOI 10.1007/s10208-004-0118-6
A. Murua. Métodos simplécticos desarrollables en P-series. Ph.D. thesis, Univ. Valladolid, 1994.
J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian problems, Applied Mathematics and Mathematical Computation, vol. 7, Chapman & Hall, London, 1994. MR 1270017
A. Zanna. A note on the implicit midpoint rule and the Euler equations for the rigid body. Private communication, 2005.