Two-dimensional supersonic flows out of rocket nozzles are one of three flow types: over-, perfectly-, or under-expanded. In under-expanded flows, an expansion fan is centered at the top and bottom tip of the rocket nozzle. When the waves from the expansion wave cross through the centerline and intersect the free boundary, the waves are reflected as compression waves. For higher exit-to-ambient pressure ratios, the compression waves coalesce and eventually form a barrel shock. The purpose of this study was to use the Method of Characteristics (MOC), a mathematical procedure for solving hyperbolic partial differential equations, to simulate the formation of the barrel shock. A MOC code was developed in the Python programming language to accomplish task. Results of the MOC code compared favorably with CFD results.