Energy conservation by
St\"ormer-type numerical integrators
E. Hairer
and
Ch. Lubich
Abstract.
For the numerical solution of second order,
highly oscillatory differential equations we study
a class of symmetric methods that includes the St\"ormer/Verlet method, the trapezoidal rule and the Numerov method.
We consider Hamiltonian systems, where high oscillations
are generated by a single frequency that is well separated from the
lower frequencies. We apply our methods
with step sizes whose product with the high
frequency is in the range of linear stability,
but is not assumed to be small.
As a main result of this paper, we show
long-time conservation of the time averages of the
total energy and the oscillatory energy, and pointwise near-preservation of
modified energies.
Key Words.
Oscillatory differential equations, long-time energy conservation,
second-order symmetric methods, St\"ormer method, Numerov method,
Fermi-Pasta-Ulam problem