An Improved WENO-Z+ Scheme (Advances in Applied Mathematics and Mechanics, 2024)
The WENO-Z+ scheme [F. Acker, R. B. de R. Borges, and B. Costa, An improved WENO-Z scheme, J. Comput. Phys., 313 (2016), pp. 726–753] with two different versions further raised the nonlinear weights with respect to the nonsmooth orless-smooth substencils, by introducing an additional term into the weights formulaof the well-validated WENO-Z scheme. These two WENO-Z+ schemes both produceless dissipative solutions than WENO-JS and WENO-Z. However, the recommendedone which achieves superior resolutions in the high-frequency-wave regions fails to recover the designed order of accuracy where there exists a critical point, while the otherone which obtains the designed order of accuracy at or near critical points is unstablenear discontinuities. In the present study, we find that the WENO-Z+ schemes over-amplify the contributions from less-smooth substencils through their additional terms,and hence their improvements of both stability and resolution have been greatly hindered. Then, we develop improved WENO-Z+ schemes by making a set of modifications to the additional terms to avoid the over-amplification of the contributions fromless-smooth substencils. The proposed schemes, denoted as WENO-IZ+, maintain thesame convergence properties as the corresponding WENO-Z+ schemes. Numericalexamples confirm that the new schemes are much more stable near discontinuitiesand far less dissipative in the region with high-frequency waves than the WENO-Z+schemes. In addition, improved results have been obtained for one-dimensional linearadvection problems, especially over long output times. The excellent performance ofthe new schemes is also demonstrated in the simulations of 1D and 2D Euler equationtest cases.