  We present 5/2- and 7/2-order $L^2$-accurate randomized Runge-Kutta-Nystr\"om
methods to approximate the Hamiltonian flow underlying various non-reversible
Markov chain Monte Carlo chains including unadjusted Hamiltonian Monte Carlo
and unadjusted kinetic Langevin chains. Quantitative 5/2-order $L^2$-accuracy
upper bounds are provided under gradient and Hessian Lipschitz assumptions on
the potential energy function. The superior complexity of the corresponding
Markov chains is numerically demonstrated for a selection of `well-behaved',
high-dimensional target distributions.
