This document summarizes a paper presented at the 23rd UK Conference of the Association for Computational Mechanics in Engineering. The paper presents a stabilized finite element method for solving compressible flow problems on moving domains. It extends the SUPG formulation to higher-order approximations in space and time using a second-order generalized alpha method. This formulation satisfies the geometric conservation law and has advantages including decoupling flow and geometry, implicit time integration, reduced computational cost with higher-order approximations, and preservation of flow effects over long times. Test problems demonstrate the method's accuracy and robustness for Burgers' equation and the Euler equations on static and moving domains.