Large area ratio nozzles that are employed for space propulsion applications are not meant to be tested at the ground-level conditions because of the flow conditions that occurs in their divergent section due to their low exhaust pressures. Therefore, it is desired to perform the experimental evaluation at high-altitude to evaluate the performance of the said nozzles. Generally, a high-altitude test facility, consisting of a supersonic exhaust diffuser, is generally employed. In thispaper, a second-throat exhaust diffuser has been numerically investigated to predict its minimum starting pressure, and to understand the flow physics in the diffuser using Computational Fluid Dynamics (CFD). Numerical computation of the flow field in the diffuser system is done for the two cases: 1) when the nozzle and the diffuser system are initially evacuated to a low pressure, and 2) when the nozzle and the diffuser system are initially at ambient pressure (1 bar). Simulations have been carried out for cold flow situation (Ȗ=1.4) over a range of nozzle inlet stagnation pressures. Numerical results compare favorably with the theoretical and experimental results and provide adequate insight to the flow physics and internal shock structures formed in thediffuser in addition to minimum starting pressure, diffuser wall pressure and vacuum thrust.