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

Invited Talk

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Control of parasitic oscillations in linear multistep methods

E. Hairer

Due to the presence of parasitic roots in symmetric linear multistep methods, the numerical solution of differential equations gives rise to non-physical oscillations. Although these oscillations have a small amplitude in the beginning, they can grow exponentially with time and soon dominate the error in the numerical approximation. Certain symmetric multistep methods for second order differential equations have the feature that these oscillations remain bounded and small (below the discretization error of the smooth solution) over very long time intervals. We extend former results for second order Hamiltonian equations to systems with holonomic constraints (index 3 differential-algebraic equations). The technique of proof is backward error analysis combined with modulated Fourier expansions. The presented results have been obtained in collaboration with Christian Lubich and Paola Console.

[1] E. Hairer and C. Lubich, Symmetric multistep methods over long times, Numer. Math. 97 (2004), pp. 699-723.
[2] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. 2nd edition. Springer Verlag, Berlin, Heidelberg, 2006.
[3] P. Console, E. Hairer and C. Lubich, Symmetric multistep methods for constrained Hamiltonian systems, to appear in Numer. Math. (2013).

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