SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Contributed Talk

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Long-term stability of multi-value methods for ordinary differential equations

R. D'Ambrosio and E. Hairer

Abstract
The recent literature regarding geometric numerical integration of ordinary differential equations has given special emphasis to the employ of multi-value methods: in particular, some authors have addressed their efforts to the construction of general linear methods with the aim of achieving an excellent long-time behavior for the integration of Hamiltonian systems. In this talk, a backward error analysis is presented, which permits to get sharp estimates for the parasitic solution components and for the error in the Hamiltonian. For carefully constructed methods (symmetric and zero growth parameters) the error in the parasitic components typically grows like $h^{p+4}\exp (h^2Lt)$, where $p$ is the order of the method, and $L$ depends on the problem and on the coefficients of the method. This is confirmed by numerical experiments.

Bibliography
[1] R. D'Ambrosio and E. Hairer, Long-term stability of multi-value methods for ordinary differential equations, submitted for publication (2013).

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