Due to the potential risks that space weather (SW) events associated with solar-wind disturbances pose on modern technology and infrastructure, there has been increasing interest in physics-based forecasts of the solar wind and related phenomena, such as coronal mass ejections. Computational models of heliospheric space plasmas and space weather are generally based on the equations of ideal magnetohydrodynamics (MHD) that describe the conservation of plasma mass, momentum, and energy, as well as the time evolution of the magnetic field. Over the last few decades significant effort has been devoted to the development of efficient numerical schemes for solving the ideal MHD equations, especially in the context of space plasmas. More recently, there has been increasing interest in incorporating observational data within SW simulations via data assimilation (DA) to produce improved space weather forecasts. While the use of DA methods is a mature field that has proved to be vastly successful in meteorological applications, its use has seen limited application in heliospheric space weather forecasting, or MHD modeling in general. In this study, the results of the assimilation of synthetic plasma observations in one-dimensional ideal MHD initial value problems are considered. DA methods are generally divided into two families of approaches: variational and sequential methods. Both categories of approaches are examined here with assimilation results presented for the 4DVar and Ensemble Kalman Filter (EnKF) methods, respectively. Observing system simulation experiments are performed and the simulation errors obtained using the 4DVar and EnKF methods, as well as, without the use of DA are compared. The sensitivity of error reduction to temporal and spatial observation availability is explored, and the computational costs of each method are reported. Finally, the challenges associated with extensions to 3D models are discussed.