Are extensions to continuum formulations for solving ﬂuid dynamic problems in the transition-to-rareﬁed regimes viable alternatives to particle methods? It is well known that for increasingly rareﬁed ﬂow ﬁelds, the predictions from continuum formulation, such as the Navier-Stokes equations lose accuracy. These inaccuracies are attributed primarily to the linear approximations of the stress and heat ﬂux terms in the Navier-Stokes equations. The inclusion of higher-order terms, such as Burnett or highorder moment equations, could improve the predictive capabilities of such continuum formulations, but there has been limited success in the shock structure calculations, especially for the high Mach number case. Here, after reformulating the viscosity and heat conduction coeﬃcients appropriate for the rareﬁed ﬂow regime, we will show that the Navier-Stokes-type continuum formulation may still be properly used. The equations with generalization of the dissipative coeﬃcients based on the closed solution of the Bhatnagar-Gross-Krook (BGK) model of the Boltzmann equation, are solved using the gas-kinetic numerical scheme. This paper concentrates on the non-equilibrium shock structure calculations for both monatomic and diatomic gases. The Landau-Teller-Jeans relaxation model for the rotational energy is used to evaluate the quantitative diﬀerence between the translational and rotational temperatures inside the shock layer. Variations of shear stress, heat ﬂux, temperatures, and densities in the internal structure of the shock waves are compared with, (a) existing theoretical solutions of the Boltzmann solution, (b) existing numerical predictions of the direct simulation Monte Carlo (DSMC) method, and (c) available experimental measurements. The present continuum formulation for calculating the shock structures for monatomic and diatomic gases in the Mach number range of 1.2 to 12.9 is found to be satisfactory.