In this work the stability and accuracy of the one-dimensional single-population Lattice Boltzmann Method (LBM) is studied in its application to compressible flow regimes. A linear stability analysis framework using the von Neumann method is developed, framed in terms of flow regime parameters. In order to validate this analysis and study the behavior of errors with respect to different simulation parameters, two academic test cases are presented. A shock tube configuration is used to assess the shock capturing capabilities. A high subsonic convergent-divergent nozzle is used to evaluate the error dependence on grid size. This test case requires the inclusion of an additional source term to the Boltzmann equation, and a methodology is proposed for the implementation of physically motivated source terms. The proposed numerical method is shown to be accurate, matching theoretical error trends. Through stability analysis, bounds for the simulation parameters resulting in a stable simulation are given, which can inform best practices in the application of LBM to these regimes.