  Numerical evolution of time-dependent differential equations via explicit
Runge-Kutta or Taylor methods typically fails to preserve symmetries of a
system. It is known that there exists no numerical integration method that in
general preserves both the energy and the symplectic structure of a Hamiltonian
system. One is thus normally forced to make a choice. Nevertheless, a symmetric
integration formula, obtained by Lanczos-Dyche via two-point Taylor expansion
(or Hermite interpolation), is shown here to preserve both energy as well as
symplectic structure for linear systems. This formula shares similarities with
the Euler-Maclaurin formula, but is superconvergent rather than asymptotically
convergent. For partial differential equations, the resulting evolution methods
are unconditionally stable, i.e, not subject to a Courant-Friedrichs-Lewy
limit. Although generally implicit, these methods become explicit for linear
systems.
