Splitting schemes for second order approximations of piecewise-deterministic Markov processes

Numerical approximations of piecewise-deterministic Markov processes based on splitting schemes are introduced, together with their Metropolis-adjusted versions. The unadjusted schemes are shown to have a weak error of order two in the step size in a general framework. Focusing then on unadjusted schemes based on the Bouncy Particle and Zig-Zag samplers, we provide conditions ensuring ergodicity and consider the expansion of the invariant measure in terms of the step size. The dependency of the leading term in this expansion in terms of the refreshment rate, depending of the splitting order, is analyzed. Finally, we present numerical experiments on Gaussian targets, a Bayesian imaging inverse problem and a system of interacting particles.