Both the Bouncy Particle Sampler (BPS) and the Discrete Bouncy Particle Sampler (DBPS) are non-reversible Markov chain Monte Carlo algorithms whose action can be visualised in terms of a particle moving with a fixed-magnitude velocity. Both algorithms include an occasional step where the particle `bounces' off a hyperplane which is tangent to the gradient of the target density, making the BPS rejection-free and allowing the DBPS to propose relatively large jumps whilst maintaining a high acceptance rate. Analogously to the concatenation of leapfrog steps in HMC, we describe an algorithm which omits the straight-line movement of the BPS and DBPS and, instead, at each iteration concatenates several discrete `bounces' to provide a proposal which is on almost the same target contour as the starting point, producing a large proposed move
with a high acceptance probability. Combined with a separate kernel designed for moving between contours, an explicit bouncing scheme which takes account of the local Hessian at each bounce point ensures that the proposal respects the local geometry of the target, and leads to an efficient, skew-reversible MCMC algorithm.