Ernst Hairer.

Title: Symmetric integrators - when do they conserve energy ?

Abstract:

The flow of a Hamiltonian differential equation exactly preserves the energy
for all times, but numerical solutions obtained with classical integration methods
typically show a linear drift in the energy.
Backward error analysis explains that symplectic methods nearly conserve
energy over exponentially long time intervals, and it was believed for a long
time that the same is true for symmetric methods applied to reversible
differential equations.

This talk discusses the long-time behaviour of symmetric integration methods.
It is illustrated and explained that the following situations may occur:

a) energy error is bounded and small over very long time:
   this is the case if a symmetric reversible method is applied to an
   integrable reversible ordinary differential equation; if the method
   is conjugate to a symplectic integrator; and in some exceptional
   cases where the modified differential equation has a first integral
   close to the Hamiltonian.

b) linear error grow: this is the typical situation, if there is no symmetry
   in the problem, and if the integrator is just symmetric and reversible,
   and does not have further geometric properties.

c) the numerical error in the energy behaves like a random walk and
   therefore grows like the square root of time: this may happen if a symmetric
   method is applied to a chaotic differential equation whose solution is
   ergodic with the presence of symmetries in the invariant measure.

Reference: Hairer, Lubich, Wanner, Geometric Numerical Integration,
   2nd edition, Springer Verlag 2006.
Related articles can be found on the homepage
   http://www.unige.ch/~hairer/preprints.html