One viable approach to the study of haemodynamics is to numerically model this flow behavior in normal and stenosed arteries. The blood is either treated as Newtonian or non-Newtonian fluid and the flow is assumed to be pulsating, while the arteries can be modeled by constricted tubes with rigid or elastic wall. Such a task involves formulation and development of a numerical method that could at least handle pulsating flow of Newtonian and non-Newtonian fluid through tubes with and without constrictions where the boundary is assumed to be inelastic or elastic. As a first attempt, the present paper explores and develops a time-accurate finite difference lattice Boltzmann method (FDLBM) equipped with an immersed boundary (IB) scheme to simulate pulsating flow in constricted tube with rigid walls at different Reynolds numbers. The unsteady flow simulations using a time-accurate FDLBM/IB numerical scheme is validated against theoretical solutions and other known numerical data. In the process, the performance of the time-accurate FDLBM/IB for a model blood flow problem and the ease with which the no-slip boundary condition can be correctly implemented is successfully demonstrated.