Improved high-order high-resolution energy stable weighted essentially non-oscillatory plus scheme for shock/vortex problems

In the present study, a high-order high-resolution energy stable weighted essentially non-oscillatory plus (ESWENO-P) scheme was developed by improving the weighting function in the energy stable weighted essentially non-oscillatory (ESWENO) scheme. First, the weighting function was modified by introducing a new user-defined parameter. Then, the fine-tuning term in the weighted essentially non-oscillatory-Z plus (WENO-ZP) scheme was adopted to the modified weighting function. A parametric study was conducted to determine the value of the user-defined parameter included in the fine-tuning term. To validate the accuracy and the resolution of the present ESWENO-P scheme, well known one- and two-dimensional benchmark flow problems involving shocks and vortices were tested. It was found that, compared to the ESWENO, the present ESWENO-P scheme is more stable near strong discontinuities. It was also observed that, compared to the WENO-ZP, the present ESWENO-P scheme was less sensitive to the user-defined parameter included in the fine-tuning term. Overall, compared to the existing weighted essentially non-oscillatory (WENO) family schemes, the present ESWENO-P scheme consistently resolves flows with high resolution without much computational overhead.