Numerical simulations are carried out for a highly under-expanded hydrogen jet resulting from an accidental release of high-pressure hydrogen into the atmospheric environment. The predictions are made using two independent CFD codes, namely CFX and KIVA. The KIVA code has been substantially modified by the present authors to enable large eddy simulation (LES). It employs a one-equation sub-grid scale (SGS) turbulence model, which solves the SGS kinetic energy equation to allow for more relaxed equilibrium requirement and to facilitate high fidelity LES calculations with relatively coarser grids. Instead of using the widely accepted pseudo-source approach, the complex shock structures resulting from the high under-expansion is numerically resolved in a small computational domain above the jet exit. The computed results are used as initial conditions for the subsequent hydrogen jet simulation. The predictions provide insight into the shock structure and the subsequent jet development. Such knowledge is valuable for studying the ignition characteristics of high-pressure hydrogen jets in the safety context.