A Lie group generalized- method with improved
accuracy
Martin Arnold (Martin Luther University
Halle-Wittenberg), Johannes Gerstmayr, Stefan Holzinger
Lie group integrators solve initial value problems for (ordinary)
differential equations on manifolds with Lie group structure. For
one-step methods, the application of classical ODE time integration
methods to a locally defined equivalent ODE in terms of local
coordinates has become a quasi-standard. These local coordinates are
elements of the corresponding Lie algebra. They are mapped by the
exponential map or by the Cayley map to the Lie group itself. For
typical fields of application, there are closed form expressions that
allow to evaluate these coordinate maps and the right hand side of the
locally defined equivalent ODE efficiently.
For multi-step methods and for the generalized- method with its subsidiary
variables, the situation is more complex since frequent
re-parametrizations of the manifold need to be avoided. As a practical
consequence, the corresponding Lie group methods suffer from extra local
error terms that may, however, be eliminated by appropriate correction
terms (V. Wieloch, M. Arnold: BDF integrators for mechanical systems
on Lie groups, NUMDIFF-15, 2018). Recently, these modified Lie
group integrators have been interpreted in terms of time derivatives of
the local coordinates. In that way, the accuracy of simulation results
was substantially improved (S. Holzinger, M. Arnold, J. Gerstmayr:
Improving the accuracy of Newmark-based time integration
methods, IMSD 2024, June 2024).
In the present paper, we analyse local and global errors of these
modified generalized-
methods, discuss some implementation issues and present numerical test
results that illustrate the improved accuracy.
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