  In this paper, explicit stable integrators based on symplectic and contact
geometries are proposed for a non-autonomous ordinarily differential equation
(ODE) found in improving convergence rate of Nesterov's accelerated gradient
method. Symplectic geometry is known to be suitable for describing Hamiltonian
mechanics, and contact geometry is known as an odd-dimensional counterpart of
symplectic geometry. Moreover, a procedure, called symplectization, is a known
way to construct a symplectic manifold from a contact manifold, yielding
Hamiltonian systems from contact ones. It is found in this paper that a
previously investigated non-autonomous ODE can be written as a contact
Hamiltonian system. Then, by symplectization of a non-autonomous contact
Hamiltonian vector field expressing the non-autonomous ODE, novel symplectic
integrators are derived. Because the proposed symplectic integrators preserve
hidden symplectic and contact structures in the ODE, they should be more stable
than the Runge-Kutta method. Numerical experiments demonstrate that, as
expected, the second-order symplectic integrator is stable and high convergence
rates are achieved.
