Understanding the properties of grain boundaries in polycrystalline superconductors is essential for optimizing their critical current density. Here, we provide computational simulations of 2D Josephson junctions (JJs) in low magnetic fields using time–dependent Ginzburg–Landau theory, since they can be considered a proxy for a grain boundary between two grains. We present data for junctions with a wide range of superconducting electrodes of different Ginzburg–Landau parameter (κ) values and geometries, as well as normal barriers with different strengths of pair–breaking — characterized by the thickness of the junction and the junction condensation parameter (α˜n). We describe our results using analytic solutions, and hence provide a detailed description of Josephson junctions in low fields up to that required for a single fluxon to penetrate the junction.