A high-order weighted compact nonlinear scheme for compressible flows

This paper presents a weighted compact nonlinear scheme (WCNS) with high-order accuracy. The WCNS has the flexibility to select a numerical flux and is easily applied to general coordinates. In this study, the specific substencils of the original targeted essentially non-oscillatory (TENO) scheme, which is effective against discontinuities with a high-order interpolation, are combined with WCNS. The accuracy of the interpolation can be easily extended, and the expressions of the fifth, seventh, ninth, and eleventh-order accuracies are provided in this paper. For two types of weight coefficients (WCNS-JS and WCNS-T), the present scheme is investigated using a various benchmark-test problems including strong shock and high-frequency waves. Within substencils, the WCNS-JS can capture strong discontinuities; however, it cannot resolve small-scale fluctuations because of the dissipation effect. On the contrary, the WCNS-T with a TENO weighing coefficient can capture strong discontinuities and high-frequency waves because high-order accuracy works effectively on the premise that a suitable grid resolution is satisfied.