Initalizing and Stabilizing Variational Multistep Algorithms For Modeling Dynamical Systems
Authors: C.L. Ellison, J.W. Burby, J.M. Finn, H. Qin and W.M. Tang
Abstract:
Backward error initialization and parasitic mode control are well-suited for use in algorithms that arise from
a discrete variational principle on phase-space dynamics. Dynamical systems described by degenerate Lagrangians,
such as those occurring in phase-space action principles, lead to variational multistep algorithms
for the integration of rst-order di erential equations. As multistep algorithms, an initialization procedure
must be chosen and the stability of parasitic modes assessed. The conventional selection of initial conditions
using accurate one-step methods does not yield the best numerical performance for smoothness and stability.
Instead, backward error initialization identi es a set of initial conditions that minimize the amplitude of
undesirable parasitic modes. This issue is especially important in the context of structure-preserving multistep
algorithms where numerical damping of the parasitic modes would violate the conservation properties.
In the presence of growing parasitic modes, the algorithm may also be periodically re-initialized to prevent
the undesired mode from reaching large amplitude. Numerical examples of variational multistep algorithms
are presented in which the backward error initialized trajectories outperform those initialized using highly
accurate approximations of the true solution.
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Submitted to: Journal of Computational Physics (January 2014)
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