Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions

Ludwig Gauckler
Date and time
Workshop - Multiscale numerical methods for differential equations

We consider trigonometric and modified trigonometric
integrators applied to oscillatory Hamiltonian differential equations
with several constant high frequencies. Examples are the impulse method
which coincides with the method of Deuflhard, the mollified impulse
method, the St\"ormer--Verlet method and an IMEX method.

For the exact solution of these equations, one has exact conservation of
the total energy and long-time near-conservation of the oscillatory
energy. For the considered numerical methods, we will show in the talk
near-conservation of the total and oscillatory energies over long time
scales that cover arbitrary negative powers of the step size.

A main issue in such long-time results are resonances between the
frequencies and resonances between the frequencies and the time
step-size. The presented results require non-resonance conditions
between the time step-size and the frequencies, but in contrast to
previous results they do not require any non-resonance conditions among
the frequencies.

This is joint work with Ernst Hairer (Geneva) and Christian Lubich (Tübingen).


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