Numerical simulations have been performed to investigate the ignition of combustible mixtures using jets (stationary and traversing) of chemically active gas. A stationary two-dimensional constant-volume combustor (CVC) is considered, with long aspect ratio similar to a wave-rotor combustor, which is ignited by a stationary or traversing hot jet from a rotatable pre-combustion chamber. The CVC initially contains a quiescent stoichiometric mixture of fuel and air, either cold (298 K) or relatively hot (514 K); the prechamber uses rich fuel-air mixtures. Different jet traverse speeds are used that span a range of jet vortex behavior and fluid-dynamic and chemical time scales. Ignition of methane-air mixture in the CVC chamber is investigated using a 21-species reaction mechanism (DRM19). Combustion initiated by hot-jet ignition includes complex jet penetration and entrainment, vortex dynamics, ignition of a quiescent combustible mixture, subsequent flame and pressure-wave propagation, and their interactions. In the present work, adaptive mesh refinement (AMR) and multi-zone techniques were employed to resolve jet and vortex structures, ignition kernels, propagating turbulent flames, pressure waves, and flame-wave interactions accurately. Jet evolution, vortex structure and mixing behavior, ignition delay time, effect of shock flame interaction for stationary centered jets, near-wall jets, and various traversing jets are investigated to help assess the relative importance of physical mixing and chemical kinetic processes. Two different combustor initial temperatures and different assumptions about jet chemical activity are considered. For methane ignition at both initial temperatures, it is observed that shock-flame interaction is seen to significantly increase the overall reaction rate due to baroclinic vorticity generation driving flame area increase, stirring, and mixing. In general, the combustion process appears to be significantly controlled by fluid dynamic processes of the jet and the shock-flame interactions, with weaker influence of the initial temperature and initial chemical activity of the jet gases.