Randomized Runge-Kutta-Nyström

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.