A new scale-invariant hybrid WENO scheme for steady Euler and Navier-Stokes equations

The steady-state convergence property of prevalent weighted essentially non-oscillatory (WENO) schemes usually relies on the sensitivity parameter. On the one hand, it is discovered that relatively large sensitivity parameter is conducive to attaining the steady-state convergence, however, large sensitivity parameter may result in some oscillations in the numerical solutions. On the other hand, relatively small sensitivity parameter can prevent some non-physical oscillations, but this choice may degrade the accuracy of WENO schemes. To address this issue, we design a fifth-order scale-invariant WENO scheme for steady problems to drag the residual of numerical solutions into machine-zero level. A new effective smoothness detector is introduced in this scheme, then the whole computational domain is classified into smooth, non-smooth and transition regions accordingly. The optimal fifth-order linear reconstruction is used in smooth region, the mixed WENO reconstruction is utilized in non-smooth region, and a interpolation technique is adapted in transition region to ensure robust steady-state convergence. In particular, the essentially non-oscillatory (ENO) property of the mixed reconstruction is verified by investigating the smoothness indicator of the mixed polynomial. Moreover, the scheme further achieves the scale-invariant property in theory, and maintains the fifth-order accuracy regardless of the order of critical points. Numerical experiments demonstrate that the scale-invariant error of this WENO scheme is close to machine zero, and the ENO property is still retained for small scale problems. What's more, the scheme is robust for the steady-state convergence across extensive benchmark examples of Euler and Navier-Stokes (NS) equations, and still displays the ENO property for the problems involving strong discontinuities.