Fluid dynamic computations are performed to predict the post-shock gas properties for high enthalpy earth entry conditions experimented at the NASA EAST facility. The inherent disparity of length- and time-scales, and a highly stiff reaction dynamics necessitate the use of implicit solvers to make such simulations tractable. The fully coupled, linearized system of equations for EAST flow is solved in the US3D flow solver using the approximate iterative implicit data-parellal line relaxation (DPLR) or full matrix data parellal point-relaxation (FMDP) methods, and with a preconditioned GMRes based linear system solver. These calculations are done in both, a fixed- and a moving-frame of reference. Substantial improvements are realized in the computational cost and accuracy by solving the flow in a moving-frame traveling with the shock. The FMDP method with sufficient sub-iterations during the off-diagonal relaxation process gives accuracy similar to the GM-Res based solver, while significantly reducing the computation cost. A carefully designed non-uniform grid is employed with a periodic tracking of the shock front location, and the grid resoultion in the computational domain is such that the regions near strong discontinuities and large viscous gradients are well resolved at all times. Additionally, the solver framework in US3D is improvized to mimic high-temperature kinetics and mitigate numerical instabilities accumulated during the lengthy time-integration process for this high speed shock propagation in a long tube. The catalytic recombination of the ions by the cold wall along with accurately linearized mass diffusion fluxes ensures a realistic behavior of the ionized species and enhances the numerical stability of the linear system. The 2D axisymmetric EAST shock tube flow is hence computed in a time-accurate, yet computationally inexpensive, manner. The numerical solution successfully captures the key flow physics and post-shock electron density predictions are consistent with the experimental observations.