In this study, a mathematical model for a two-dimensional pulsatile blood flow through a stenosed artery in the presence of a magnetic field and body acceleration is simulated. The artery is assumed to be an elastic cylindrical tube and stenosis's geometry is considered time-dependent to provide resemblance to the in-vivo situations. The blood flow has been assumed to be non-linear, incompressible, and fully developed. Moreover, due to the pulsatile pressure gradient arising from systematic functioning of the heart and the body acceleration, the blood flow has been assumed unsteady. To facilitate the application of the finite difference scheme in solving the governing equations, a coordinate transformation is employed, transforming the stenosed artery into a rectangular artery. The nonlinear momentum and the continuity equations under suitable initial and boundary conditions can be numerically solved using the Crank-Nicolson scheme. The blood flow characteristics such as the velocity profile, the volumetric flow rate, and the resistance to flow are obtained and the effects of the severity of the stenosis and the magnetic field on these flow characteristics are discussed. Besides the blood flow characteristics through elastic arteries have been compared with the rigid ones.