In this work we deal with the numerical solution of the fluid-structure interaction problem arising in the haemodynamic environment. In particular, we consider BDF and Newmark time discretization schemes, and we study different methods for the treatment of the fluid-structure interface position, focusing on partitioned algorithms for the prescription of the continuity conditions at the fluid-structure interface. We consider explicit and implicit algorithms, and new hybrid methods. We study numerically the performances and the accuracy of these schemes, highlighting the best solutions for haemodynamic applications. We also study numerically their convergence properties with respect to time discretization, by introducing an analytical test case.