A Filtered Embedded Weighted Compact Non-Linear Scheme for Hyperbolic Conservation Law

Abstract

In situations where a wide range of flow scales are involved, the non-linear scheme should be capable of both shock capturing and low-dissipation. Most of the existing Weighted Compact Non-linear Schemes (WCNS) are too dissipative and incapable of achieving fourth-order for the two smooth stencils located on the same side of a discontinuity due to the weight deviations and the defect of the weighting strategy. In this paper, a novel filtered embedded WCNS is introduced for complex flow simulations involving both shock and small-scale structures. To overcome the above deficiency of existing WCNS, a pre-discrete mapping function is proposed to filter the weight deviation out and amend the inappropriate weights to ideal weights in smooth regions. Meanwhile, the embedded process is also implemented by this function, which is utilized to improve the resolution of shock capturing in certain discontinuity distributions. The pre-discrete mapping function is also extended to the WENO framework. The approximate-dispersion-relation analysis indicates that the scheme with the mapping function has lower dispersion and dissipation error than the WCNS-JS, WCNS-Z, and WCNS-T schemes. Numerical results show that WCNS with the new non-linear weights captures discontinuities sharply without obvious oscillation, has a higher resolution than other non-linear schemes, and has an obvious advantage in capturing small-scale structures.