Weighted Compact and Non- compact Scheme for Shock Tube and Shock Entropy Interaction

[Abstract] In this paper, we introduce a new type of high order shock capturing schemes – uniform weighted compact and non-compact scheme (UWCNC) or simply XJL scheme developed by Xie, Jiang and Liu. This new scheme is based on the feature of discrete data sets instead of the physics. The fundamental task of CFD is to provide an accurate approximation of derivatives for a given discrete data set. The data are first normalized then measured by so called " smoothness ". According to the smoothness, the set is divided into three regions: smooth, oscillatory, and non-differentiable (shock) regions. The strategy of this new scheme is to achieve spectral-like resolution and high order of accuracy by using central weighted compact scheme in smooth and oscillatory regions and use the non-compact scheme to cross the shock to capture shocks sharply without oscillation. In a 6th order one parameter family of the compact schemes (Lele, 1992), we turn the control parameter to 1/3, keeping the exact formulation of the 6th order weighted compact scheme and turn it to zero gradually when approaching the shock, which makes the scheme non-compact. Besides the WENO weights, there is only one additional control parameter, which we call smoothness function. In this new uniform weighted compact-non compact scheme, a sixth-order weighted compact scheme and a corresponding fourth-order weighted non-compact scheme are combined following this basic idea. Numerical tests show that the new scheme has captured the 1-D shock sharply without non-physical oscillation and obtained much higher resolution for 1-D shock-entropy interaction than the 5th order WENO scheme.