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The accurate prediction of aerodynamic drag on satellites orbiting in the upper atmosphere is critical to the operational success of modern space technologies, such as satellite-based communication or navigation systems, which have become increasingly popular in the last few years due to the deployment of constellations of satellites in low-Earth orbit. As a result, physics-based models of the ionosphere and thermosphere have emerged as a necessary tool for the prediction of atmospheric outputs under highly variable space weather conditions. This paper proposes a high-fidelity approach for physics-based space weather modeling based on the solution of the Navier-Stokes equations using a high-order discontinuous Galerkin method, combined with a matrix-free strategy suitable for high-performance computing on GPU architectures. The approach consists of a thermospheric model that describes a chemically frozen neutral atmosphere in non-hydrostatic equilibrium driven by the external excitation of the Sun. A novel set of variables is considered to treat the low densities present in the upper atmosphere and to accommodate the wide range of scales present in the problem. At the same time, and unlike most existing approaches, radial and angular directions are treated in a non-segregated approach. The study presents a set of numerical examples that demonstrate the accuracy of the approximation and validate the current approach against observational data along a satellite orbit, including estimates of established empirical and physics-based models of the ionosphere-thermosphere system. Finally, a 1D radial derivation of the physics-based model is presented and utilized for conducting a parametric study of the main thermal quantities under various solar conditions., Comment: 26 pages, 13 figures, 5 tables