Accurate numerical prediction of the post-shock gas properties in high enthalpy environments is challenging and computationally expensive due to the strongly coupled and extremely stiff flow physics. Utmost care is required to devise a numerical method to solve the governing equations in a time-accurate and thermodynamically consistent manner. The current study examines various approaches to compute unsteady high speed flow in the NASA Ames EAST shock-tube and demonstrates their vulnerability to produce incorrect or spurious results. The model advection-diffusion equation is used to gain thorough understanding of the numerical artifacts that arise when solving coupled systems of equations with strong hyperbolic character. The model equation reveals the need of the implicit system of equations to be solved accurately for proper shock-tracking. Through careful analysis, a time-accurate numerical solution for flow in the EAST facility is obtained and a robust and expedient method of solution is presented.