A New ϵ-Adaptive Algorithm for Improving Weighted Compact Nonlinear Scheme with Applications

To improve the resolution and accuracy of the high-order weighted compact nonlinear scheme (WCNS), a new ϵ-adaptive algorithm based on local smoothness indicators is proposed. The new algorithm introduces a high-order global smoothness indicator to adjust the value of ϵ according to the local flow characteristics. Specifically, the algorithm increases ϵ in smooth regions, which can help cover up the disparity in smoothness indicators of sub-stencils and make the nonlinear scheme approach the background linear scheme. As a result, optimal order accuracy can be achieved in smooth regions, including critical points. While near discontinuities, the algorithm decreases ϵ, thereby strengthening the stencil selection mechanism and further attenuating spurious oscillations. Meanwhile, the new algorithm makes nonlinear schemes scale-invariant of flow variables. The results of approximate dispersion relation (ADR) show that the new algorithm can greatly reduce spectral errors of nonlinear schemes in the medium and low wavenumber range without inducing instability. Numerical results indicate that the new algorithm can significantly improve resolution of small-scale structures and suppress numerical oscillations near discontinuities with only a minor increment in computational cost.