Application of central differencing and low-dissipation weights in a weighted compact nonlinear scheme

This paper proposes WCNS-CU-Z, a weighted compact nonlinear scheme, that incorporates adapted central difference and low-dissipative weights together with concepts of the adaptive central-upwind sixth-order weighted essentially non-oscillatory scheme (WENO-CU) and WENO-Z schemes. The newly developed WCNS-CU-Z is a high-resolution scheme, because interpolation of this scheme employs a central stencil constructed by upwind and downwind stencils. The smoothness indicator of the downwind stencil is calculated using the entire central stencil, and the downwind stencil is stopped around the discontinuity for stability. Moreover, interpolation of the sixth-order WCNS-CU-Z exhibits sufficient accuracy in the smooth region through use of low-dissipative weights. The sixth-order WCNS-CU-Zs are implemented with a robust linear difference formulation (R-WCNS-CU6-Z), and the resolution and robustness of this scheme were evaluated. These evaluations showed that R-WCNS-CU6-Z is capable of achieving a higher resolution than the seventh-order classical robust weighted compact nonlinear scheme and can provide a crisp result in terms of discontinuity. Among the schemes tested, R-WCNS-CU6-Z has been shown to be robust, and variable interpolation type R-WCNS-CU6-Z (R-WCNS-CU6-Z-V) provides a stable computation by modifying the first-order interpolation when negative density or negative pressure arises after nonlinear interpolation.