A computational approach is developed to simulate high enthalpy shock propagation under strong nonequilibrium conditions, with application to atmospheric entry flows in shock tubes. Numerical simulations are performed for 6-11 km/s atmospheric entry shock experiments conducted in the Electric Arc Shock Tube (EAST) facility at NASA Ames Research Center. The simulation strategy developed here involves simplifications of the EAST facility and optimization of computational methods to make numerical calculations tractable. Shock tube flow evolution is mathematically represented by a system of time- varying, strongly hyperbolic partial differential equations (PDEs) with stiff source-terms. An 11-species weakly ionized air model is used to capture the real-gas behavior with reaction rates from Park (1990) and Huo (2015). Species mass- diffusion fluxes are calculated as- suming a self-consistent effective binary diffusion (SCEBD) in conjunction with Gupta-Yos (1990) collision-integral data. Starting from simple one-dimensional wave propagation problems, the computational modeling is systematically improved for two-dimensional axisymmetric simulations of EAST flow, pertinent to Earth and Titan entry conditions. The unsteady, linearized system of conservation laws is solved in a moving-frame of reference with active shock tracking to reduce the computational cost by three orders of magnitude relative to that of fixed-frame calculations on a uniform grid. This approach enables first-ever predictions of gas behavior at EAST test-section in a time-accurate yet computationally feasible manner. Numerically computed shock-deceleration and post-shock electron number densities are compared against EAST measurments. CFD data qualitatively captures the nonequilibrium gas behavior observed in the experiments despite some discrepancies due to factors such as modeling assumptions, simplifications of the experimental process, limitations of gas physical and chemistry models under high-enthalpy strong nonequilibrium conditions, etc. Shock deceleration in CFD is about 40 to 100 times smaller and post-shock electron densities are lower than that measured in EAST. CFD data is further used to understand shock start- up process at different shock speeds. Useful quantities such as boundary layer thickness and growth rate, and radial variation of gas properties are also studied, these are crucial for radiative heating calculations but cannot be directly measured in the experiments.