  We introduce novel Markov chain Monte Carlo (MCMC) algorithms based on
numerical approximations of piecewise-deterministic Markov processes obtained
with the framework of splitting schemes. We present unadjusted as well as
adjusted algorithms, for which the asymptotic bias due to the discretisation
error is removed applying a non-reversible Metropolis-Hastings filter. In a
general framework we demonstrate that the unadjusted schemes have weak error of
second order in the step size, while typically maintaining a computational cost
of only one gradient evaluation of the negative log-target function per
iteration. Focusing then on unadjusted schemes based on the Bouncy Particle and
Zig-Zag samplers, we provide conditions ensuring geometric ergodicity and
consider the expansion of the invariant measure in terms of the step size. We
analyse the dependence of the leading term in this expansion on the refreshment
rate and on the structure of the splitting scheme, giving a guideline on which
structure is best. Finally, we illustrate the competitiveness of our samplers
with numerical experiments on a Bayesian imaging inverse problem and a system
of interacting particles.
