Time-symmetry, symplecticity and stability of Euler-Maclaurin and Lanczos-Dyche integration

Understand

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.