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

Contributed Talk

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An a priori analysis of parasitism for G-symplectic multivalue methods

A. Hill and T. Norton

Abstract
We consider a G-symplectic multivalue method approximating a Hamiltonian differential equation. We say that parasitic instability is controlled if it grows like $\exp(\nu nh^2)$ for some $\nu>0$. A linear analysis indicates that controlled instability is almost inevitable. Nevertheless, such methods, which include the Leapfrog method, are worth pursuing because of their geometric properties and relatively low implementation costs. Necessary conditions on the method coefficients are obtained to ensure that the instability is controlled for a linear constant coefficient problem. The desirability of additional non-stiff assumptions, on the time-step and the ODE, is inferred from a linear non-autonomous example. Under these and further assumptions, we prove in the nonlinear case that the growth of the leading parasitic term is controlled.

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