One of the crucial issues of computational fluid dynamics is how to discretize the viscous terms accurately. Recently, an attractive and viable alternative numerical method for solving the compressible Navier–Stokes equations is proposed. The first-order hyperbolic system (FOHS) with reconstructed discontinuous Galerkin (rDG) method was first proposed to solve advection–diffusion model equations and then extend to compressible Navier–Stokes equations. For the model advection–diffusion equation, the proposed method is reliable, accurate, efficient, and robust, benefiting from FOHS and rDG methods. To implement the method of compressible Navier–Stokes equations, the gradients of density, velocity, and temperature are introduced as auxiliary variables. Numerical experiments demonstrate that the developed HNS + rDG methods are able to achieve the designed order of accuracy for both primary variables and their gradients.

A first-order hyperbolic reformulation of the Navier-Stokes equations provides an attractive alternative to conventional solvers by treating flow gradients as additional solution variables, thereby avoiding explicit reconstruction of viscous derivatives. However, its application to steady supersonic and hypersonic flows remains unexplored because of the challenges associated with strong shock waves and thin boundary layers. In this work, an implicit hyperbolic Navier-Stokes method for steady supersonic and hypersonic viscous flows is presented. By reformulating the Navier-Stokes equations into a first-order hyperbolic system, the proposed approach directly evolves density, velocity, andtemperature gradients together with the conservative variables, eliminating the need for conventional viscous gradient reconstruction. To improve robustness in the presence of strong discontinuities, a gradient compression-factor strategy is incorporated.The proposed method is assessed using a series of canonical steady supersonic and hypersonic problems, ranging from one-dimensional Ma∞ = 2.0 viscous shock structure, Ma∞ = 2.0 supersonic flow past a diamond wedge, Ma∞ = 4.5 laminar flow past a flat plate to Ma∞ = 8.03 and Ma∞ = 15.0 hypersonic flow past a blunt body.Numerical results demonstrate that the proposed method accurately captures shock structures and boundary layers while maintaining good conservative properties. Compared to a conventional finite-volume Navier-Stokes solver, the new formulation provides better flow gradients through the direct evolution, and the compression-factor ensures the robustness. This work provides an alternative numerical strategy for steady high-speed compressible viscous flows.