Reconstruction-based high-order conservative central difference schemes for the compressible Navier–Stokes equations

A class of reconstruction-based high-order conservative central difference schemes (CD) is developed for solving the compressible Navier–Stokes equations in this paper. The discretization of the viscous and heat fluxes in the Navier–Stokes equations involves a two-step process, where the external first derivatives of the viscous terms are disposed in a reconstruction way, and then the interpolation operation is carried out to calculate the internal first derivatives within the same stencil. Two approaches to the interpolation implementation are discussed: one is founded on the conservative variables, while the other is based on the primitive variables. This design can maintain compactness and consistence as in the stencil of the weighted essentially non-oscillatory (WENO) schemes for the inviscid terms. Under the present framework, a sixth-order central difference scheme for the viscous terms is designed with a stencil width that falls within the range of the fifth-order WENO scheme for the inviscid terms. The accuracy for both linear and nonlinear diffusion equations are demonstrated theoretically and the spectral properties are verified via Fourier analysis. Numerous compressible viscous results validate that the present central difference schemes are high-order accurate in smooth regions, easy to implement, robust for the viscous shock simulations and computationally cost-effective.