An unequal-sized weighted compact nonlinear scheme with adaptive numerical flux for simulating compressible flows

The weighted compact nonlinear scheme (WCNS) is a widely used shock-capturing method for compressible flows, but reducing its numerical dissipation has always been a challenge. In this paper, an unequal-sized weighted essentially non-oscillatory (US-WENO) node-to-midpoint interpolation procedure combined with a novel adaptive numerical flux is proposed to enhance its simulation capability for three-dimensional inviscid and viscous compressible flows. First, inspired by the reconstruction-based US-WENO scheme, a new node-to-midpoint interpolation procedure suitable for the WCNS method was developed and named US-WCNS. Compared with the classical WCNS method, this developed method adopts a five-point and two two-point US stencils, and its linear weights do not need to be calculated (i.e., they can be arbitrarily chosen), instead of using three three-point equidistant stencils with unique linear weights that require additional calculation. Second, a novel adaptive Lax–Friedrichs (LF) numerical flux calculation method was proposed based on the recently proposed extremum properties discontinuous sensor. Compared with the commonly used global LF numerical flux, this method has smaller numerical dissipation and thus achieves higher resolution for small-scale structures. Finally, some benchmark examples are provided to verify the performance of this proposed method in terms of numerical accuracy, shock-capturing characteristics, and dissipation properties. For the three-dimensional inviscid Taylor–Green vortex problem, the kinetic energy preservation effect of the proposed method is improved by 18% compared to the classical WCNS scheme at the same grid level.