Simplified weighting formulations of weighted compact nonlinear schemes for compressible flows

Weighted compact nonlinear schemes (WCNSs) are a popular family of high-resolution shock-capturing schemes for simulating compressible flows, of which the nonlinear interpolation procedure is dominant for the performance. In this work, a simplified weighting strategy is introduced for the nonlinear interpolation procedure. Firstly, an equivalent weighting formulation of WCNS is presented by explicitly including the whole-point stencil into the set of candidate stencils. Secondly, motivated by the reorganization of WCNS, the WCNS-CU6 scheme is achieved in a more straightforward way. Thirdly, by introducing a TENO selection procedure in the framework of WCNS-CU6-Simplified, a TCNS6-Simplified scheme is proposed, the resolution of which is comparable with the excellent TENO6 scheme, while the computational cost is much lower. The simplified schemes exhibit more outstanding, at least comparable, fidelity than the original schemes, however, with superior characteristics in terms of efficiency and simplicity. A variety of benchmark test problems are studied to demonstrate the behaviour of the simplified weighting strategy.