A symplectic structure preserved by the trapezoidal rule

The trapezoidal rule has often been referred to as being symmetric or time-reversible and is therefore good for Hamiltonian systems. However, it is well-known that the trapezoidal rule is not symplectic, but is related to the mid-point rule (which is symplectic) through a coordinate transformation. In this paper, we show that the trapezoidal rule preserves a symplectic structure different from the original one by O(h²). The ideas in this paper also motivate us to apply Richardson's extrapolation to the trapezoidal rule. Numerical results show that the extrapolated trapezoidal rule preserves the Hamiltonian up to O(h⁴).