A turbulent combustion model based on the Linear Eddy Model for Large Eddy Simulation (LEM- LES) is currently proposed to study self-ignition events of rapidly expanding hydrogen jets. The model is a one-dimensional treatment of a diffusion-reaction system within each multi-dimensional LES cell. This reduces the expense of solving a complete multi-dimensional problem while preserving micro-scale hotspots and their effects on ignition. The current approach features a Lagrangian description of fluid particles on the sub-grid for increased accuracy. Also, Adaptive Mesh Refinement (AMR) is implemented for increased computational efficiency. In this paper, the model is validated for various inviscid laminar 1-D mixing and ignition problems, shock tube problems, flames, and detonations.