Recent Results on the Improved WENO-Z+ Scheme
The WENO-Z scheme is known to achieve less dissipative results than the classical WENO scheme, especially in problems involving both shocks and smooth structures. In Acker et al. (J Comput Phys 313:726–753, 2016), the cause of the improved results of WENO-Z was shown to be its comparatively higher weights on less-smooth substencils. This knowledge was exploited to develop the fifth-order WENO-Z+ scheme, which generalizes WENO-Z by including an extra term for increasing the weights of less-smooth substencils even further. The new scheme WENO-Z+ was shown to achieve even better results than WENO-Z, while keeping the same numerical robustness. In this study, the third- and seventh-order versions of the WENO-Z+ scheme are presented and discussed. The preliminary numerical results make evident that the approach used by WENO-Z+ is also sound for orders other than 5.
The author would like to thank Profs. Wai-Sun Don and Guus Jacobs for organizing together the minisymposium of High-order Methods at ICOSAHOM 2016; Prof. Chi-Wang Shu and the Division of Applied Mathematics of Brown University, where part of this research was done; and the organizers of ICOSAHOM 2016, especially Prof. Marco Bittencourt, for all the support.
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